Advanced Ramsey Dimensional Analysis

Abstract

We propose a Ramsey approach to the dimensional analysis of physical systems, which complements the seminal Buckingham theorem. Dimensionless constants describing a given physical system are represented as vertices of a graph, referred to as a dimensions graph. Two vertices are connected by an aqua-colored edge if they share at least one common dimensional physical quantity and by a brown edge if they do not. In this way, a bi-colored complete Ramsey graph is obtained. The relations introduced between the vertices of the dimensions graph are non-transitive. According to the Ramsey theorem, a monochromatic triangle must necessarily appear in a dimensions graph constructed from six vertices, regardless of the order of the vertices. Mantel–Turán analysis is applied to study these graphs. The proposed Ramsey approach is extended to graphs constructed from fundamental physical constants. A physical interpretation of the Ramsey analysis of dimensions graphs is suggested. A generalization of the proposed Ramsey scheme to multi-colored Ramsey graphs is also discussed, along with an extension to infinite sets of dimensionless constants. The introduced dimensions graphs are invariant under rotations of reference frames, but they are sensitive to Galilean and Lorentz transformations. The coloring of the dimensions graph is independent of the chosen system of units. The number of vertices in a dimensions graph is relativistically invariant and independent of the system of units.

1. Introduction

Dimensional analysis, introduced by Joseph Fourier and James Clerk Maxwell and developed by Lord Rayleigh, remains one of the most powerful, important, and elegant approaches for the qualitative analysis and solving of physical problems [1,2,3,4,5]. Dimensional analysis rests on an extremely simple and intuitively clear, yet fundamental, principle: physical laws are independent of the arbitrary choice of basic units of measurement. Despite its simplicity, dimensional analysis yields remarkably deep, general, and important results, contributing, among other things, to advances in the theories of gravity and turbulence [3,6]. Recently, the principle of dimensional invariance has been incorporated into a two-level machine learning framework designed to automatically identify dominant dimensionless numbers and governing laws—including scaling relations and differential equations—from limited experimental data [4].

One of the fundamental results of dimensional analysis is the famous Buckingham $\pi$-theorem [4,5,6,7], which addresses an arbitrary physical system, described by a number of physical quantities/parameters, denoted as $Q_i(i=1,\dots ,n)$, and their ratios, denoted as $r_j\left(j=1,\dots l\right).$ The most general form of the physical equations describing the behavior of this system is provided by Equation (1):

$$f\left(Q_1,Q_2\dots Q_n,r_1,\dots ,r_l\right)=0$$

(1)

If it is assumed that the ratios do not vary during the evolution of the system described by the equation, i.e., the system remains similar to itself during the changes in quantities/parameters, $Q_i(i=1,\dots ,n)$, Equation (1) is reduced to Equation (2):

$$f\left(Q_1,Q_2\dots Q_n\right)=0$$

(2)

The seminal Buckingham theorem (which is also known as the “$\pi$-theorem”) states that Equation (2) may be re-shaped in terms of $\zeta$ dimensionless parameters, $\zeta =n-k=n-\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(D\right)$, where k is the maximal number of values possessing independent dimensions among n values of parameters $Q_i$, denoted $Q_m,\left(m=1,\dots ,k\right),$ and D is the dimension matrix [7,8,9,10,11]. The columns of the dimension matrix are the physical variables; the rows, in turn, are the fundamental dimensions. The parameters $Q_m(i=1,\dots ,k)$ are considered dimensionally independent if the dimensions of any of $Q_m$ cannot be expressed through the dimensions of other quantities from that subset [7,8,9]. Although named for Edgar Buckingham, the “π-theorem” was first proved by the French mathematician Joseph Bertrand in 1878 [10]. From a purely mathematical point of view, the Buckingham theorem is rooted in the rank–nullity theorem of linear algebra [11]. Our paper presents a new approach to dimensional analysis that is complementary to the Buckingham theorem. This approach is based on the Ramsey theorem, applied in its graph form.

One of the most advanced fields of modern mathematics is graph theory, which has been extensively developed in the last decades [12,13,14,15,16,17,18,19]. A mathematical graph is a structure used to represent pairwise relationships between objects. In simple terms, a graph represents a collection of objects, together with the set of pairwise relationships between them [12,13,14,15,16,17,18,19]. It consists of vertices (or nodes), which serve as the fundamental units of the graph, and edges (or links), which denote the connections between these vertices [12,13,14,15,16,17,18,19]. A particular class of graphs, known as colored graphs, assigns colors to certain elements of the graph—typically to its vertices or edges [12,13,14,15,16,17,18,19].

A classical result in the theory of colored graphs is the Ramsey theorem, which states that for any pair of positive integers, r and s, there exists a smallest integer, $R\left(r,s\right)$, called the Ramsey number, such that in any aqua–brown edge coloring of a complete graph, $K_n$, with $n\ge R\left(r,s\right)$ vertices, there exists either an aqua-colored clique of size r (i.e., a set of r vertices where every pair is connected by an aqua edge) or a brown-colored clique of size s (i.e., a set of s vertices where every pair is connected by a brown edge) [18,19]. In particular, if we have a bi-colored, complete graph built from six vertices, it will inevitably contain at least one mono-colored triangle, or, in other words, $R\left(\mathrm{3,3}\right)=6.$ Only a restricted set of Ramsey numbers is known, and this is due to the fact that coloring problems encode extremely high-dimensional combinatorial configurations [19]. The study of colored graphs is a rapidly progressing field of mathematics [20,21]. Modern trends in the field of colored graphs are surveyed in [20,21].

This paper is organized as follows:

(i)

A mathematical procedure leading to the construction of dimensions graphs is introduced;

(ii)

Illustrative examples from hydrodynamics are provided;

(iii)

Dimensions graphs containing six vertices are analyzed, and the application of the Ramsey theorem is demonstrated, along with a discussion of the physical meaning of the corresponding Ramsey graphs;

(iv)

The behavior of dimensions graphs under Galilean and Lorentz transformations is examined;

(v)

Dimensions graphs arising from fundamental physical constants are introduced;

(vi)

The Shannon entropy of dimensions graphs is evaluated;

(vii)

An extension of the Ramsey approach to multi-colored dimensions graphs is presented;

(viii)

The discussion and conclusions summarize the research and outline directions for future investigations.

2. Results

2.1. Graph Analysis of Dimensional Problems

Let us introduce the procedure enabling the conversion of the dimensional analysis of the physical problem into a bi-colored, complete graph. We exemplify our approach with the compressible fluid/gas flow in a nozzle [22,23,24]. This problem is described with three dimensionless constants ($\pi$-groups), namely, the Mach number, Ma; the Reynolds number, Re; and the specific heat ratio, $\gamma$, i.e., the triad of quantities, supplied by Equation (3):

$$Ma={\displaystyle \frac{u}{c}};Re={\displaystyle \frac{uL}{\upsilon }};\gamma ={\displaystyle \frac{c_p}{c_v}},$$

(3)

where u is the velocity of flow; c is the speed of sound; L is the characteristic spatial scale; and $\nu$, $c_p, \mathrm{a}\mathrm{n}\mathrm{d} c_v$ represent, respectively, the kinematic viscosity and the thermal capacities of the fluid or gas at constant pressure and constant volume. This problem perfectly illustrates the Buckingham theorem. Indeed, the full list of relevant physical quantities is $Q_i(i=1,\dots ,6)=\left\{u,c,L,\nu ,c_p,c_v\right\}$, and the set of dimensionally independent values is $Q_m(m=1,\dots ,3)=\left\{u,L,c_p\right\}$. Thus, $\mathsf{\zeta }=n-k=6-3=3$ dimensionless $\pi$-groups will describe the problem. These dimensionless Buckingham $\pi$-groups are listed in Equation (3). Now, we introduce the mathematical procedure enabling the conversion of the dimensionless groups and relations between them into a bi-colored graph. The dimensionless constants, i.e., $Ma,Re,$ and $\gamma$, serve as the vertices of the graph, as shown in Figure 1.

The vertices are connected by the aqua link when they contain at least one common physical value. Different physical values possessing the same dimensions (u and c) are considered distinct. The Mach number, Ma, and the Reynolds number, Re, contain a common physical value for both of them, which is the characteristic velocity, u. These vertices are connected by the aqua-colored link, as depicted in Inset A of Figure 1. In contrast, the Mach number, $Ma$, and the specific heat ratio, $\gamma$, do not contain any common physical quantities. These vertices are connected by the brown link, as shown in Inset B of Figure 1.

The kinematic viscosity may be expressed as $\upsilon ={\displaystyle \frac{\eta }{\rho }}$, where $\eta$ and $\rho$ are the viscosity and density of the liquid, respectively. Substitution of $\upsilon ={\displaystyle \frac{\eta }{\rho }}$ does not change the color of the aqua link $\left(Ma,Re\right).$ Now, let us discuss the coloring of the link $\left(Ma,\gamma \right)$ in more detail. We assume that $c_p\ne c_p\left(u,c\right)$ and $c_v\ne c_v\left(u,c\right)$. This justifies the brown coloring of the link $\left(Ma,\gamma \right)$.

Now, we present the complete, bi-colored graph emerging from the dimensional analysis of the compressible fluid flow in a nozzle. This graph, depicted in Figure 2, is addressed by the Ramsey theorem [12,13]. Recall that a complete graph is a type of graph in mathematics (specifically in graph theory) in which every pair of distinct vertices is connected by a unique edge [12,13].

Figure 2 illustrates a very important idea: the property “to be connected by a brown” link is not transitive. Vertices Ma and $\gamma$ are connected by a brown link, and vertices $\gamma$ and Re are connected by a brown link; this does not imply that the vertices, Ma and Re, are necessarily connected by the brown link.

The triad of the dimensionless numbers, $\left\{Ma,Re,\gamma \right\}$, may be time-dependent. In contrast, the coloring of the graph remains constant during the evolution of the flow, as long as it is described by the same triad of dimensionless quantities.

Figure 2 depicts the bi-colored, complete graph addressed by the Ramsey theorem. It should be mentioned that mono-colored triangles built on the dimensionless constants, seen as vertices of the graph, are possible. Consider oscillatory phenomena occurring in incompressible liquids described by the triad of dimensionless constants, supplied by Equation (4):

$$Ma={\displaystyle \frac{u}{c}};Re={\displaystyle \frac{uL}{\upsilon }};St={\displaystyle \frac{fL}{u}},$$

(4)

where St is the Strouhal number, and f is the oscillation or vortex shedding frequency. The full list of relevant physical values is $Q_i(i=1,\dots ,5)=\left\{u,c,f,L,\nu \right\}$, and the set of dimensionally independent values is $Q_m(m=\mathrm{1,2})=\left\{f,L,\right\}$. Thus, $\mathsf{\zeta }=n-k=5-2=3$ dimensionless $\pi$-groups will describe the problem. However, in this case, the graph corresponding to the problem will be monochromatic.

We refer to the graphs depicted in Figure 2 and Figure 3 as dimensions graphs. The dimensions graph depicted in Figure 3 is monochromatic since all of the vertices (dimensionless numbers) contain the velocity of the flow, denoted as u; i.e., the triad of the dimensionless numbers is velocity-dependent. Furthermore, recalling $\upsilon ={\displaystyle \frac{\eta }{\rho }}$, the aqua-coloring of the graph remains unchanged.

It is evident that the property “to be connected by an aqua-link” is not necessarily transitive. In other words, if groups ${\pi }_i$ and ${\pi }_k$ are connected by an aqua-colored link, and groups ${\pi }_k$ and ${\pi }_l$ are also connected by an aqua-colored link, it does not necessarily imply that groups ${\pi }_i$ and ${\pi }_l$ are connected by such an edge. The non-transitivity of the connection is crucial for the future application of the Ramsey theorem to the analysis of dimensions graphs. An additional example of the monochromatic aqua-colored dimensions graph is supplied in Appendix A.

It should be emphasized that the coloring of the dimensions graph is independent of the chosen system of units when we restrict ourselves to the $LMT$ systems of units. If we use the systems of units in which $c=1$ or $h=1$ or both of the fundamental units, i.e., the velocity of light and the Planck constant equal to unity, the coloring of the dimensions graph may be different from that supplied by the LMT systems (see Section 2.7).

2.2. Dimensions Graphs Built from Six Vertices and the Ramsey Theorem

Now consider the flow of an incompressible, viscous, heat-conducting fluid under general/non-simplified conditions. This complex hydrodynamic system is governed by the Navier–Stokes equations (for momentum), the continuity equation (for mass conservation), and the energy equation (for heat transfer) [22,23,24]. This problem is typically quantified with six dimensionless numbers, listed in Equation (5):

$$Ma={\displaystyle \frac{u}{c}};Re={\displaystyle \frac{uL}{\upsilon }};St={\displaystyle \frac{fL}{u}};Pr={\displaystyle \frac{c_p\rho \nu }{\kappa }};Fr={\displaystyle \frac{u}{\sqrt{gL}}};Ec={\displaystyle \frac{u^2}{c_p\mathsf{\Delta }\mathrm{T}}},$$

(5)

where Pr, Fr, and Ec are the Prandtl, Froude, and Eckert numbers, correspondingly; $\rho$ and $\upsilon$ are the density and kinematic viscosity of the liquid; $\mathsf{\Delta }T$ is the difference between the wall temperature and the local temperature; and g is gravity. The Prandtl number, Pr, quantifies the ratio of momentum diffusivity to the thermal diffusivity, $\kappa$; the Eckert number, Ec, expresses the relationship between the flow’s kinetic energy and the boundary layer enthalpy difference and is used to characterize heat transfer dissipation. The Froude number, Fr, defines the ratio of the flow inertia to gravity, g. Let us build the dimensions graph: dimensionless $\pi$-groups listed in Equation (5) serve as the vertices of the graph, depicted in Figure 4.

The full list of relevant physical values is $Q_i(i=1,\dots ,10)=\left\{u,c,f,L,\nu ,T,\rho ,\kappa ,c_p,g\right\}$. The set of dimensionally independent values is $Q_m(m=1,\dots 4)=\left\{f,L,T,\rho \right\}$. Thus, according to the Buckingham theorem, $\mathsf{\zeta }=n-k=10-4=6$ dimensionless $\pi$-groups/vertices appear in the dimensions graph, as shown in Figure 4.

The graph containing six $\pi -\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{s}/$vertices is of particular interest. According to the Ramsey theorem, the Ramsey number is $R\left(\mathrm{3,3}\right)=6$ [17,18] (see the Introduction section). Thus, the bi-colored, complete graph containing six vertices inevitably includes at least one mono-colored triangle. We have a total of $\left(\begin{array}{c}5\\ 3\end{array}\right)=10$ aqua-colored triangles in the graph. Indeed, triangles $\left\{Ma,Fr,Ec\right\}$, $\left\{Pr,Re,Ec\right\}$, $\left\{Ma,Re.Ec\right\},\left\{Ma,Re,Fr\right\}$, $\left\{Ma,St,Ec\right\}$,$\left\{Ec,St,Re\right\}$,$\left\{Ma,Re,St\right\},\left\{Re,St,Fr\right\},$ and $\left\{Ma,St,Fr\right\}$ are monochromatic, aqua-colored ones. The monochromatic triangle will be present in any complete, bi-colored dimensions graph built from six vertices. Moreover, for the given problem, changing the order of the vertices of the dimension graph does not change the distribution of the monochromatic triangles. When we permute the vertices of the dimensions graph, we are simply relabeling them. This does not affect the edge colors; it just changes their names. Therefore, the structure of the dimensions graph remains the same, and so does the number and type of monochromatic triangles. Thus, we demonstrate the following theorem:

Theorem 1.

Consider a dimensions graph built from six vertices. Every vertex represents the dimensionless number ($\pi$-group) built from dimensional physical quantities. The vertices are connected by the aqua-colored link when they contain at least one dimensional physical quantity common to both vertices. The vertices are connected by a brown link when they do not contain any physical quantity common to both vertices. The dimensions graph inevitably contains at least one mono-colored triangle, whether aqua- or brown-colored.

Straightforward application of the Ramey theorem becomes possible due to the fact that the relations “to be connected by the aqua-colored link” and “to be connected by the brown link” are both non-transitive. This is illustrated by triangles $\left\{Pr,Ec,Fr\right\}$, $\left\{Pr,Ec,Ma\right\}$, and $\left\{Pr,St,Ma\right\}$. The Ramsey numbers for the graphs in which the vertices are related with transitive relations are different from the “regular non-transitive Ramsey numbers” [25].

The coloring of the graph remains constant during the evolution of the flow as long as it is described by the same set of dimensionless quantities $\pi -\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{s}$. In our analysis of the graph, depicted in Figure 4, we implicitly assumed that the expressions describing the dimensionless $\pi -\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{s}$ are irreducible to other functional dependencies between the physical variables. This may be wrong for specific physical systems. Consider the edge $\left(Ma,Pr\right),$ taken as brown in Figure 4. If we consider $c=\sqrt{{\displaystyle \frac{K}{\rho }}}$ (where $K$ is the isentropic bulk modulus), which is generally correct for liquids, it will appear as aqua-colored. Thus, the coloring of the dimension graph is not unique. However, according to the Ramsey theorem, the bi-colored dimensions graph will always contain at least one monochromatic triangle, regardless of the coloring of the graph.

2.3. The Physical Meaning of the Ramsey Analysis of the Dimensions Graph

We now address the physical meaning of the Ramsey analysis of the dimensions graph. Ramsey theory, simply speaking, asserts that if some set of objects is large enough, the prescribed patterns are guaranteed to appear. In our case, the Ramsey theorem states that the monochromatic triangle will necessarily appear in any dimensions graph built from six vertices. This surprising result does not follow from the Buckingham theorem. Thus, it is complementary to the Buckingham theorem. If the monochromatic triangle is aqua-colored, it means that the triad of physical phenomena, depending on the same physical value/pair of the same physical values, is necessarily present in the graph. Consider, for example, the aqua-colored triangle, $\left\{Ma,Re,St\right\}.$ The triad of its vertices contains the velocity of the flow, u. Thus, we have in our problem a triad of flow–velocity-dependent physical phenomena. The same is true for the triangles $\left\{Ma,Fr,Ec\right\} \mathrm{a}\mathrm{n}\mathrm{d} \left\{Re,St.Ec\right\}.$ We do not recognize the brown triangles in the dimensions graph, shown in Figure 4. This means that under the flow of a compressible, viscous, heat-conducting fluid, there is no triad of independent physical phenomena. Indeed, if the monochromatic triangle is brown-colored, we have a triangle representing a triad of independent physical events. Regrettably, the Ramsey theorem does not predict the exact color of the monochromatic triangle to be present in the graph built from six vertices [18,19]. However, involving the Mantel–Turán Extremal Theorem, to be discussed in Section 2.6, sometimes enables exact prediction of the appearance of monochromatic polygons in a graph.

The Buckingham $\pi$-theorem guarantees the existence of dimensionless groups, but it does not uniquely specify which groups to choose. The set of dimensionless groups is not unique, and there can be many mathematically valid, but physically different, choices. Our choice of $\pi$-groups should be physically justified, and this choice has been defined, until now, by our physical intuition. So, various dimensions graphs related to the same physical problem are possible, and their analysis is based on our physical understanding of the problem.

2.4. Dimensions Graphs and Galilean and Lorentz Transformations

Let us put forward the following fundamental question: what transformations will prevent the dimensions graph from changing? From a purely mathematical point of view, the dimensions of physical values are described by the “dimension matrix”, D [7,8,9,10,11]. Recall that the columns of the dimensions graph are the physical variables; each column provides the exponents of the base units in that variable. Rows, in turn, are the fundamental/base dimensions; each row lists the exponent of that base dimension across all variables. Changes in units within the same dimension system, $M^{\prime }\to \alpha M;L^{\prime }\to \beta L;T^{\prime }=\gamma T$, maintains the dimension matrix D unchanged. Rotations of frames (orthogonal transformations in space) and time translations also keep the matrix, D, untouched.

We are in the realm of physics, so the fundamental question is formulated as follows: are dimensions graphs sensitive to Galilean or Lorentz transformations? Consider the problem described by a single $\pi -\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}$, namely, the Reynolds number, $Re={\displaystyle \frac{uL}{\upsilon }}={\displaystyle \frac{uL\rho }{\eta }}$, where $\nu$ and $\eta$ are the kinematic and dynamic viscosity of the liquid, respectively, and $\rho$ is the density. We consider the $M,L,T$ system of units. The dimensional matrix, D, for the variables $\left(\rho ,u,L,\eta \right)$ is as follows: columns are variables, and rows are $M,L,T$.

$$D=\left(\begin{array}{c}1\\ -3\\ 0\end{array}\begin{array}{c}0\\ 1\\ -1\end{array}\begin{array}{c}0\\ 1\\ 0\end{array}\begin{array}{c}1\\ -1\\ -1\end{array}\right)$$

(6)

A Galilean change in inertial frame (adding a constant velocity, U, to all particle velocities) simply replaces the numerical value, $u\to u-U$, with the velocity variable. That operation does not change the exponents of M, L, T for the variable u, so the column for u remains (0, 1, −1). The same is true for other variables if the base-dimension basis is held fixed (the set of primitive dimensions and their interpretation are unchanged). Lorentz boost changes the numerical values of the observables (length contraction and time dilation), and density picks up a Lorentz factor, $\gamma ={\displaystyle \frac{1}{\sqrt{1-{\left(\frac{U}{c}\right)}^2}}}$, but those Lorentz factors are dimensionless functions of velocities U and c. Multiplying a quantity by a dimensionless factor does not change its dimensional exponents. Thus, the row of exponents for that quantity in the dimensional matrix stays the same. Hence, the dimensional matrix, D, is unchanged. For example, the numerical value of the Reynolds number, Re, can change (you can make $u=0$ in a co-moving frame so that $Re=0$). This is a frame-dependent number, not a change in dimensional structure. The same is true for Lorentz transformations. Even in relativistic contexts, the dimensions of the variables $\left(\rho ,u,L,\eta \right)$ remain the same. Again, Lorentz boosts change the coordinate components numerically but do not change the bookkeeping exponents unless we change the unit conventions. Thus, frame boosts (Galilean or Lorentz) only change the numerical components of variables, not their dimensional exponents—so they do not change the dimensional matrix if we keep the same variable list and the same base dimensions. Thus, we come to a very important theorem:

Theorem 2.

The dimension matrix, D, is the Galilean and Lorentz invariant.

• Hence, the number of the vertices of the dimensions graph, which coincides with the number of the dimensionless $\pi$-groups, is also relativistically invariant.

Thus, it seems that the coloring of the dimensions graph is insensitive to Galilean and Lorentz transformations. However, this conclusion is wrong. Both Galilean and Lorentz transformations will introduce additional dimensional parameters, namely, the velocities of frames and the velocity of light, c. Thus, the coloring of the dimension graph may be sensitive to Lorentz and Galilean transformations.

Generally speaking, the most general transformations of frames that leave the dimensions graph untouched are those that preserve the dimensional structure of physical quantities, i.e., transformations belonging to the automorphism group of the dimensional graph. The study of this group is out of the scope of the presented paper.

2.5. Dimensions Graph and Fundamental Physical Constants

Now, we apply the developed approach to the analysis of the dimensionless numbers constructed from fundamental dimensional physical constants. Dimensional fundamental physical constants are physical quantities whose values cannot be derived from other physical quantities within the framework of existing physical theories. We disconnect from the specific physical problem and address dimensionless numbers uniting basic physical constants. The set of addressed dimensionless numbers include

$$\alpha ={\displaystyle \frac{e_e^2}{4\pi {\epsilon }_0\hslash c}};{\alpha }_G={\displaystyle \frac{G{m}_p^2}{\hslash c}};N={\displaystyle \frac{e_p^2}{4\pi {\epsilon }_{0}G{m}_P^2}};\mu ={\displaystyle \frac{m_p}{m_e}};\xi ={\displaystyle \frac{e_p}{e_e}};\theta ={\displaystyle \frac{\mathsf{\Lambda }\hslash G}{c^3}}=\mathsf{\Lambda }{l}_p^2,$$

(7)

where $e_e$ and $m_e$ are the electrical charge and mass of the electron, $e_p$ and $m_p$ are the electrical charge and mass of the proton, G and $\Lambda$ are the gravitational and cosmological constants, $\hslash$ and c are the Planck constant and light velocity in a vacuum, and $l_p=\sqrt{{\displaystyle \frac{\hslash G}{c^3}}}$ is the Planck length. In Equation (6), $\alpha \cong {\displaystyle \frac{1}{137}}$ is the fine structure constant, which quantifies the strength of the electromagnetic interaction between elementary charged particles; ${\alpha }_G\cong 5.9\times {10}^{-39}$ is the gravitation coupling constant, which measures the strength of gravity between two protons, analogous to the fine-structure constant for gravity; $\mu ={\displaystyle \frac{m_p}{m_e}}\cong 1836$ compares the mass of the proton to that of the electron; the dimensionless constant, N, relates electric interactions between protons to gravitational interaction $N\cong 1.23\times {10}^{36};$ the dimensional constant $\xi ={\displaystyle \frac{e_p}{e_e}}=1$ compares the charge of the proton to that of the electron; and $\theta ={\displaystyle \frac{\mathsf{\Lambda }\hslash G}{c^3}}=\mathsf{\Lambda }{l}_p^2=2.87\times {10}^{-122}$. This extremely small, dimensionless number is often cited in discussions of the cosmological constant problem, indicating how unnaturally small the observed value is relative to Planck-scale expectations. Let us represent these numbers with a dimensions graph, as depicted in Figure 5.

The scroll of dimensional physical quantities now includes

$$Q_i\left(i=1,\dots ,10\right)=\left\{m_p,m_e,e_p,e_e,G,\hslash ,c,G,\mathsf{\Lambda },{\epsilon }_0\right\}.$$

(8)

The set of dimensionally independent values is $Q_m(m=1,\dots 4)=\left\{m_p,e_p,c,\hslash \right\}$. Thus, $\mathsf{\zeta }=n-k=10-4=6$ dimensionless groups/vertices appear in the bi-colored, complete dimensions graph, as depicted in Figure 5. Recall that we use the SI system of units; this explains the appearance of ${\epsilon }_0$ in the list of dimensional values. The Ramsey theorem states that at least one mono-colored triangle will be necessarily present in the graph. Indeed, we recognize aqua-colored triangles $\left\{\alpha ,\xi ,N\right\}$, $\left\{{\alpha }_G,\theta ,N\right\}$, $\left\{{\alpha }_G,\theta ,\alpha \right\}$, and $\left\{{\alpha }_G,\mu ,N\right\}$ and one brown triangle $\left\{\theta ,\mu ,\xi \right\}$ in the graph, as shown in Figure 5. Let us analyze the mono-colored triangles. Quantities ${\epsilon }_0,e_e,e_p$ make triangle $\left\{\alpha ,\xi ,N\right\}$ aqua-colored. This means that just the electrical phenomena unite the triad of dimensionless constants, $\left\{\alpha ,\xi ,N\right\}$. Now consider the aqua-colored triangle, $\left\{{\alpha }_G,\theta ,N\right\}$. Just the gravitational constant, G, makes it aqua-monochromatic. In other words, gravitation unites the triad of dimensionless constants, $\left\{{\alpha }_G,\theta ,N\right\}.$ We now address the aqua-colored triangle, $\left\{{\alpha }_G,\theta ,\alpha \right\}.$ A pair of fundamental constants, G and c, make it monochromatic. Consider aqua-colored triangle $\left\{{\alpha }_G,\mu ,N\right\}.$ Just the proton mass $m_p$ makes it monochromatic, i.e., inertia/gravity joins the effects described by the triad of dimensionless constants $\left\{{\alpha }_G,\mu ,N\right\}.$

Now address the brown triangle $\left\{\theta ,\mu ,\xi \right\}.$ The vertices of this triangle do not contain common physical quantities. This means that the triad of dimensionless values, $\left\{\theta ,\mu ,\xi \right\}$, describes physical phenomena that are different in their physical nature. The dimensions graph is insensitive to Galilean transformations however sensitive to Lorentz transformations. It should be emphasized that the vertices of the graph are built from the fundamental physical quantities only. Thus, no functional dependence between these values is adopted, and this is in contrast to the dimensions graphs emerging from the “regular” physical problems, as discussed in Section 2.2. Fundamental physical constants cannot be expressed (within our current theory) as a function of other, more basic physical constants. Thus, the coloring of the dimensions graph, in this case, is unambiguous.

Again, the dimensions graph emerging from the basic physical interactions is not unique [26]. A diversity of fundamental constants relating basic physical interactions has been suggested [26,27]. In our future investigations, we plan to address transformations of dimensions graphs under various mathematical procedures.

2.6. Use of the Mantel–Turán Theorem for the Analysis of Dimension Graphs

The Ramsey theorem does not predict the exact color of the triangle to be necessarily present in the complete, bi-colored graph, built from six vertices, such as that depicted in Figure 5. Thus, we apply advanced graph–theoretical analysis, based on the Mantel–Turán theorem [12,13]. The total number of edges in the graph built of the six vertices is given by

$$N_{tot}=N_a+N_b=\left(\begin{array}{c}6\\ 2\end{array}\right)=15,$$

(9)

where $N_a$ and $N_b$ are the numbers of the aqua- and brown-colored links, respectively. These numbers are $N_a=10$ and $N_b=5$ for the graph shown in Figure 5. The Mantel–Turán Extremal Theorem predicts that when $N_a\ge 10,$ the graph will inevitably contain at least one aqua-colored triangle [12,13]. Indeed, the dimensions graph, presented in Figure 5, contains monochromatic aqua-colored triangles.

The Mantel–Turán analysis supplied an even more surprising result. Consider the dimensions graph in which $N_a\ge 10$. In this case, dimensionless quantities are forced into at least one closed dimensional loop. This corresponds to the emergence of a universal scaling relation. A closed aqua triad corresponds to three observables sharing one hidden common dimensional quantity, a latent variable. Therefore, systems with a sufficiently rich dimensional structure must develop scaling laws in the form ${\pi }_i=f\left({\pi }_j,{\pi }_k\right).$

2.7. Dependence of Dimensions Graphs on the System of Units

In physics, we often use a system of units in which $c=1$ or $h=1$ (or in which both c and h are set equal to 1, also known as the natural system of units [27]). Obviously, the coloring of the dimensions graph depends on the assumed systems of units. However, simple analysis based on linear algebra demonstrates that the number of the dimensionless $\pi -\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{s}, \zeta$, i.e., $\zeta =n-k=n-\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(D\right)$, does not depend on the system of units. Consequently, the number of vertices in the dimensions graph is independent of the adopted physical system of units. Moreover, all of the $\pi -\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{s}$ will remain dimensionless in the natural systems of units ($c=1$ or $h=1$, or both of them are equal to unity; see Appendix B). This is clear from the following reasoning. Any physical quantity, Q, may be represented as

$$Q=Q_{num}\times \left|Q\right|,$$

(10)

where $Q_{num}$ is the numerical value of the quantity and $\left|Q\right|$ is its dimension. Thus, if the quantity, Q, is transformed by changing the units, $Q\to Q^{\prime }$, Equation (11) is true, as the physical quantity itself does not change:

$$Q_{num}\times \left|Q\right|=Q_{num}^{\prime }\times \left|Q^{\prime }\right|$$

(11)

From Equation (11), it follows that a dimensionless physical quantity remains dimensionless in any system of units, and its numerical value remains unchanged. Since π-groups are dimensionless physical quantities, a dimensions graph constructed with six vertices will inevitably contain at least one monochromatic triangle, regardless of the chosen system of units. In contrast, if we use the systems of units in which $c=1$ or $h=1$ or both fundamental units are set to unity, the exact coloring of the dimensions graph may be different than that obtained using the conventional LMT systems.

2.8. Shannon Entropy of Dimensions Graphs

Ramsey’s theorem claims that if a system is large enough, then some kind of order must appear, no matter how chaotically we arrange it. It rules out the idea of “total chaos” in sufficiently large structures. More rigorously speaking, monochromatic triangles necessarily appear in bi-colored complete graphs when the number of vertices is equal to or larger than six. The question is how the order emerging in the complete graph may be quantified. It may be quantified with Shannon entropy [28]. The Shannon entropy of a bi-colored Ramsey complete graph is labeled S, and it is introduced with Equation (12):

$$S=-{\sum }_k{P}_{k}ln{P}_k,k\ge 3,$$

(12)

where $P_k$ is the fraction of monochromatic k-polygons/polygons with k sides (whether aqua- or brown-colored) in the given complete graph [28]. The sampling takes place over the entire set of monochromatic polygons, regardless of what their colors are. Let us exemplify this approach. The dimensions graph emerging from the analysis of compressible fluid flow in a nozzle, as shown in Figure 2, contains no monochromatic polygons; thus, $P_k=0,k\ge 3$. Therefore, we calculate that $S=0.$ The dimension graph describing the incompressible oscillatory flow of the liquid contains a single aqua-colored triangle. This leads to $P_3=1,S=0.$

What is the meaning of S$?$ Shannon entropy, S, is adequately interpreted as the average uncertainty of finding the monochromatic polygon (regardless of its color) within the set of monochromatic polygons appearing in the given complete bi-colored graph. The value of Shannon entropy is exhaustively defined by the distribution of polygons in the given complete graph, and it is independent of the exact shapes of the polygons.

The Shannon measure of the dimensions graph may be introduced using an alternative method with a pair of Shannon entropies, namely, $S_a$ and $S_b$. They are interpreted as follows: $S_a$ is interpreted as the average uncertainty of finding the aqua-colored polygon in the given graph, and $S_b$ is, in turn, the average uncertainty of finding the brown polygon in the same graph [16]. In this case, sampling of polygons is carried out separately from the aqua- and brown-colored subsets of convex polygons. Thus, a pair of Shannon entropies $\left(S_a,S_b\right)$ corresponds to any momenta graph. The pair of Shannon entropies is introduced as follows:

$$S_a=-{\sum }_n{P}_{na}ln{P}_{na},n\ge 3$$

(13)

$$S_b=-{\sum }_i{P}_{ib}ln{P}_{ib},i\ge 3$$

(14)

where $P_{na}$ is the fraction of monochromatic aqua-colored convex polygons with n aqua-sides or edges (aqua-colored edges), and $P_{ib}$ is the fraction of monochromatic convex brown polygons with i b-sides or edges (brown edges) in a given complete dimensions graph. Sampling of polygons is carried out separately from the aqua- and brown-colored subsets of convex polygons [28]. Thus, a pair of Shannon entropies $\left(S_a,S_b\right)$ corresponds to any momenta graph. For the graph depicted in Figure 2, we calculate $P_{na}=0$; $P_{nb}=0;n\ge 3;$ hence, $\left(S_a,S_b\right)=\left(\mathrm{0,0}\right).$ For the graph depicted in Figure 2, we calculate $P_{3a}=1$; $P_{3b}=0$. Hence $\left(S_a,S_b\right)=\left(\mathrm{0,0}\right)$.

Let us now compute the Shannon entropies for the graphs shown in Figure 4 and Figure 5. In total, with six vertices, it is possible to form 20 triangles, 15 quadrilaterals, 6 pentagons, and 1 hexagon. The total number of polygons is 42. For the graph depicted in Figure 4, we recognize the convex, monochromatic polygons listed in

Table 1. Distribution of convex, monochromatic polygons in the dimensions graph depicted in Figure 4.

Color	Triangles	Quadrilaterals	Pentagons	Hexagons
Aqua	11	7	1	0
Brown	0	0	0	0

.

Thus, with Equation (12), we calculate that $S=-\left({\displaystyle \frac{11}{19}}ln{\displaystyle \frac{11}{19}}+{\displaystyle \frac{7}{19}}ln{\displaystyle \frac{7}{19}}+{\displaystyle \frac{1}{19}}ln{\displaystyle \frac{1}{19}}\right)\cong 0.8392$. For the pair $\left(S_a,S_b\right),\mathrm{w}\mathrm{e} \mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{e}$ $S_a=-P_{3a}\ln P_{3a}-P_{4a}\ln P_{4a}-P_{5a}\ln P_{5a}=0.8392$ and $S_b=0$.

We have 11 triangles, 7 quadrilaterals, and 1 pentagon. $P_{3a},P_{4a}$ and $P_{5a}$ are ${\displaystyle \frac{11}{19}}$, ${\displaystyle \frac{7}{19}} \mathrm{a}\mathrm{n}\mathrm{d} {\displaystyle \frac{1}{19}}$, respectively. In this case, we will get $S_a=-P_{3a}\ln P_{3a}-P_{4a}\ln P_{4a}-P_{5a}\ln P_{5a}=0.8392$. As there is no brown polygon, $S_b=0$. Thus, $\left(S_a,S_b\right)=\left(0.8392;0\right).$

Similarly, for Figure 5, we establish the following data supplied in

Table 2. Distribution of convex, monochromatic polygons in the dimensions graph depicted in Figure 5.

Color	Triangles	Quadrilaterals	Pentagons	Hexagons
Aqua	6	4	1	0
Brown	1	0	0	0

.

In total, we have 12 monochromatic, convex polygons. Thus, for S, in this case, we calculate $S=-\left({\displaystyle \frac{7}{12}}ln{\displaystyle \frac{7}{12}}+{\displaystyle \frac{4}{12}}ln{\displaystyle \frac{4}{12}}+{\displaystyle \frac{1}{12}}ln{\displaystyle \frac{1}{12}}\right)\cong 0.887$. For the pair $\left(S_a,S_b\right)$ we derive, in turn, $S_a=-P_{3a}\ln P_{3a}-P_{4a}\ln P_{4a}-P_{5a}\ln P_{5a}=-\left({\displaystyle \frac{6}{11}}\right)ln\left({\displaystyle \frac{6}{11}}\right)-\left({\displaystyle \frac{4}{11}}\right)ln\left({\displaystyle \frac{4}{11}}\right)-\left({\displaystyle \frac{1}{11}}\right)ln\left({\displaystyle \frac{1}{11}}\right)=0.9178$, and $S_b=-\left({\displaystyle \frac{1}{1}}\right)ln\left({\displaystyle \frac{1}{1}}\right)=0$. Eventually, we establish $\left(S_a,S_b\right)=\left(0.9178;0\right).$

2.9. Extension of the Ramsey Approach to Multi-Colored Dimensions Graphs

The introduced Ramsey approach can be naturally generalized to multi-colored dimensions graphs, such as those depicted in Figure 6, illustrating the flow of an incompressible, viscous, heat-conducting fluid, already discussed in Section 2.1. Now, we connect two vertices with the red link when they contain at least two physical quantities common to both vertices/dimensionless groups. Indeed, the Re, St, and Fr numbers contain velocity, u, and a characteristic spatial scale, L, and they are connected by the red links, as shown in Figure 6.

The rest of the edge coloring remains untouched. Now, we recognize the $\left\{Ma,Fr,Ec\right\}$ {Ma, Re, Ec}, and $\left\{Ma,St,Ec\right\}$ aqua-colored triangles and the red-colored triangle $\left\{Re,St,Fr\right\}$ in the dimensions graph, as presented in Figure 6. The red triangle $\left\{Re,St,Fr\right\}$ confirms the triad of physical phenomena, which are both dependent on the characteristic spatial scale, L, and velocity of the flow, u. The same three-colored approach may be applied for the analysis of complete graphs constructed from fundamental physical constants, such as the graph depicted in Figure 5. It is noteworthy that $R\left(\mathrm{3,3},3\right)=17$; hence, in the three-colored, complete dimensions graph constructed from 17 vertices, there will necessarily be at least one mono-colored triangle present. The suggested coloring is also exemplified by Appendix C.

Three-colored graphs may also emerge when $\pi -\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{s}$ are functionally dependent. According to the Buckingham theorem, $\pi -\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{s}$ are independent in the dimensional (algebraic) sense: they form a basis for all dimensionless monomials. However, this does not ensure that they are independent in the functional sense; physical laws can (and often do) impose further relations among $\pi -\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{s}$, namely, ${\pi }_i=f\left({\pi }_j,\right)$ takes place. If ${\pi }_i=f\left({\pi }_j\right)$ is true, we connect ${\pi }_i$ and ${\pi }_j$ with the red link, and the three-colored Ramsey graph emerges; $R\left(\mathrm{3,3},3\right)=17$ remains valid.

2.10. Generalization of the Ramsey Approach to Infinite-Dimensions Graphs

The introduced approach may be generalized to infinite sets of dimensionless constants. Consider an infinite, yet countable, set of dimensionless constants. The constants are connected by the aqua-colored link if they contain at least one physical value common to both of them. The constants are connected by a brown link if they do not contain a physical value common to both of them. In this case, the infinite Ramsey theorem becomes applicable. Let us rigorously formulate the infinite Ramsey theorem. Let $K_{\omega }$ denote the complete colored graph on the vertex set, N. For every $\mathsf{\Gamma }>0$, if we color the edges of $K_{\omega }$ with $\mathsf{\Gamma }$ distinguishable colors, then an infinite monochromatic clique must be present [29]. An infinite monochromatic clique in a colored graph is a subset of vertices that are all pairwise-adjacent (in other words, they form a clique) and whose edges/links are all the same color in a given edge/link coloring of the given infinite graph. The infinite Ramsey theorem reformulates the well-known Dirichlet pigeonhole principle, which states that if n pigeonholes exist containing $n+1$ pigeons, one of the pigeonholes necessarily must contain at least two pigeons [29]. Thus, a monochromatic aqua-colored or brown-colored clique will necessarily appear in the infinite-dimensions graph. The infinite Ramsey theorem does not predict the exact color of the monochromatic clique to be present in the infinite-dimensions graph.

3. Discussion

We introduced the graph theory approach to the dimensional analysis of physical problems. Let us discuss the weak and problematic aspects of the suggested approach.

(i)

The notion of a dimensionless physical constant remains vague. For example, the number of space–time, $\tilde{N}$, coordinates may be considered a dimensionless fundamental physical constant [30]. A dimensionless fundamental constant is understood as a dimensionless quantity that cannot be derived from existing physical theories but is determined from a full set of experimental data. For example, within the current theory, the dimensionality of space–time, denoted $\tilde{N}$, is considered a structural property of the Universe and cannot be derived from other measured physical quantities [30]. Thus, N may be considered one of the vertices of the dimensions graph built from fundamental physical constants (see Figure 5). All edges from N to any other vertex will be brown. In graph–theoretic terms, N is a brown hub, a vertex that connects to all others of the same color. This makes N a very strong candidate for appearing in monochromatic triangles in Ramsey analysis. The situation is even more complicated if we consider the Boltzmann constant, $k_B$. The Boltzmann constant was originally introduced in 1900 by Planck in his analysis of blackbody radiation. Planck assumed that the Boltzmann constant is a fundamental constant and that when it is established for molecular motions, it will be the same for radiation phenomena [31]. However, the classical textbook of statistical physics by Landau and Lifshitz displays a different attitude toward the Boltzmann constant [32]. According to Landau and Lifshitz, as well as other respectable sources, the Boltzmann constant plays a much more modest role in physics. It is no more than a numerical coefficient transforming one unit of energy into another, similar to the numerical coefficient transforming the length unit of yards into meters. Thus, the definition of the “dimensions graph built from fundamental physical constants” becomes ambiguous, and it depends on our current understanding of the fundamental structure of nature.

(ii)

The introduced coloring may also be ambiguous due to the functional dependencies between the physical parameters appearing in the graph, as discussed in Section 2.1 and Section 2.2.

(iii)

As the number of constants increases (say, in field theories with many dimensionless couplings), the dimensions graph expands in complexity. Ramsey theory guarantees substructures, but interpreting them may become combinatorially opaque. Calculation of large Ramsey numbers remains an unsolved problem. However, asymptotic estimations for the number of monochromatic subgraphs in bi-colored graphs have already been calculated [33,34]. The asymptotic proportion of monochromatic t-cliques in any two-colouring of a complete graph constructed from n vertices, denoted as $C_t,$ is given by the Conlon formula [34]:

$$c_t\ge C^{-\left(1+o\left(1\right)\right)t^2},C\cong 2.18$$

(15)

In the limit $n\to \infty$, $c_3={\displaystyle \frac{1}{4}}$. So, in the limit $n\to \infty$, at least 25% of all triangles in the dimension graph are monochromatic, no matter how you color the edges [33].

4. Conclusions

Ramsey theory is a branch of discrete mathematics/combinatorics that studies conditions under which order must appear in sufficiently large or complex systems, and demonstrates the essential potential for physics. Its power lies in guaranteeing structure in sufficiently large systems. The immediate conclusion emerging from Ramsey theory is formulated as follows: in sufficiently large systems, complete chaos is impossible. Patterns inevitably appear in systems when their size is larger than the critical one. At least one monochromatic triangle will inevitably appear in complete, bi-colored graphs built from six vertices. In spite of this, the applications of Ramsey theory to physical problems are still rare in occurrence [35,36,37,38,39].

We apply Ramsey theory to reveal hidden patterns inherent to sets of dimensionless numbers, which are used for the qualitative analysis of physical problems. These dimensionless numbers are built from dimensional physical values, appearing in a given physical problem. The number of these dimensionless groups (known also as $\pi -\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{s})$ is established by the seminal Buckingham $\pi -\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m},$ arising from the rank–nullity theorem of linear algebra [7,8,9,10,11]. We suggest a mathematical scheme complementary to the Buckingham $\pi$ − theorem, based on the theory of complete, bi-colored graphs. Consider a given physical problem. Dimensionless $\pi -\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{s}$ qualitatively defining the system serve as the vertices of the graph. Vertices are connected with the aqua-colored link, when they contain at least one dimensional physical quantity common for both vertices. Vertices are connected, in turn, by a brown link when they do not contain any physical quantity common to both vertices. Thus, the complete, bi-colored graph, labeled as the “dimensions graph”, emerges. If this graph contains six vertices, it will inevitably contain at least one mono-colored triangle, whether aqua- or brown-colored. Thus, the triad of physically interrelated or physically independent $\pi -$ groups will inevitably appear in the dimensions graph.

Buckingham focuses on constructing individual dimensionless groups. Our approach shifts attention to the network of all such groups and the relations between them. This could reveal hidden redundancies or structural patterns in physical values.

We applied the suggested mathematical procedure, uniting Buckingham $\pi$-groups into the bi-colored complete graph, for the analysis of the specific physical problem of the flow of an incompressible, viscous, heat-conducting fluid. This problem is qualitatively characterized by six dimensionless numbers ($\pi$-groups). The appearance and physical meaning of monochromatic triangles recognized within the graph were analyzed. The dimensions graph contains five velocity-dependent $\pi$-groups; thus, $\left(\begin{array}{c}5\\ 3\end{array}\right)=10$ aqua-colored triangles are present in the graph, and our problem includes ten triads of flow–velocity-dependent physical phenomena. On the other hand, we do not recognize the brown triangles in the same dimensions graph. This means that under the flow of a compressible, viscous, heat-conducting fluid, there is no triad of independent physical phenomena.

The same mathematical scheme was applied for the analysis of the graph built from dimensionless numbers comprising fundamental physical constants. The introduced dimensions graphs are independent of the choice of the unit system. It is noteworthy that the introduced dimensions graphs are invariant under the rotations of reference frames but are sensitive to Galilean and Lorentz transformations.

The Ramsey theorem does not predict the exact color of the monochromatic polygon to be present in the bi-colored graph. Sometimes, this prediction becomes possible due to Mantel–Turán analysis of dimensions graphs [40], as exemplified in our paper. Asymptotic estimates for the number of monochromatic subgraphs in bi-colored dimensions graphs are addressed.

The suggested mathematical scheme is suitable for dimensional machine learning [4]. The orderliness of the dimensions graphs was quantified using Shannon entropy. Generalization of the Ramsey graph approach, exploited for the analysis of multi-colored and infinite-dimensions graphs, is discussed. In our future investigations, we plan to study the symmetry of dimensions graphs.