Richard M. Low^a , Ardak Kapbasov^a , Arman Kapbasov^b , Sergey Bereg^c

richard.low@sjsu.edu, ardak.kapbasov@yahoo.com, akapbasov@fb.com, sbereg@gmail.com

For various connected simple graphs G, we extend the table of diagonal graph Ramsey numbers R(G, G) in 'An Atlas of Graphs.' This is accomplished by first converting the calculation of R(G, G) into a satisfiability problem in propositional logic. Mathematical arguments and scientific computing are then used to calculate R(G, G).

Keywords: Graph Ramsey theory, diagonal Ramsey numbers

2020 Mathematics Subject Classification: 05C55

DOI: 10.5614/ejgta.2022.10.2.17

1. Introduction

In 1929, Frank Ramsey [20] established an innocuous-looking theorem in his groundbreaking paper on formal logic. Although it was not apparent at the time, Ramsey's theorem would eventually form the cornerstone of Ramsey theory, a vibrant and rich area of extremal combinatorics.

The following general question [12] is investigated in Ramsey theory.

• If a particular mathematical structure (e.g., algebraic, combinatorial, or geometric) is arbitrarily partitioned into finite many classes, what kinds of substructures must always remain intact in at least one of the classes?

Received: 29 January 2022, Revised: 2 October 2022, Accepted: 9 October 2022.

^aDepartment of Mathematics and Statistics, San Jose State University, San Jose, CA 95192, USA ^bFacebook, 1 Hacker Way, Menlo Park, CA 94025, USA

^cDepartment of Computer Science, University of Texas at Dallas, Richardson, TX 75083, USA

Over many decades, Ramsey-type questions in mathematical structures such as the integers [16], graphs, and Euclidean space have been investigated. As of this writing, a keyword search for "Ramsey" yields 2926 entries in the MathSciNet database. The interested reader is directed to [12, 13] for a comprehensive overview of Ramsey theory. For a gentle introduction to Ramsey theory, [22] is recommended.

Applications of Ramsey theory can be found in number theory, algebra, geometry, topology, set theory, logic, ergodic theory, information theory and computer science. The reader is directed to Rosta's [23] survey for a detailed exposition of some of these applications.

The reader should note that the seeds of Ramsey theory were planted even before Ramsey introduced his theorem. Soifer's [25] beautifully written book is filled with deep mathematics and also provides a rich historical context of Ramsey theory.

2. Preliminaries

The focus of this paper is on calculating new diagonal Ramsey numbers in graph Ramsey theory.

First, we recall some standard definitions and notation from graph theory. All graphs are finite, simple and connected. Any notation and terminology which are not explicitly defined in this paper can be found in [12, 27]. For a graph G with vertex set V (G) and edge set E(G), the order and size of G are defined to be |V (G)| and |E(G)|, respectively. The complete graph Kⁿ is the simple graph on n vertices, where every pair of vertices are adjacent.

In graph Ramsey theory, the following definitions and notation are used.

Definition 1. Let k ≥ 2. A r-coloring of G is a coloring of E(G), using a maximum of r colors.

Notation 1. Let G and H be simple connected graphs. If every 2-coloring of Kⁿ yields a monochromatic subgraph G or a monochromatic subgraph H in Kn, then this is denoted by Kⁿ → (G, H). If that is not the case, then the notation Kⁿ ↛ (G, H) is used.

Definition 2. The Ramsey number R(G, H) is defined to be the minimum n, where Kⁿ → (G, H).

Using graph-theoretic language, the finite version of Ramsey's theorem can be stated in the following way.

Theorem A. (Ramsey [20]). Let s, t ≥ 2. Then, there exists a minimal positive integer n such that every edge coloring of Kⁿ (using two colors) contains a monochromatic K^s or a monochromatic K^t .

Considerable work has been done in graph Ramsey theory. In addition to the calculation of Ramsey numbers in the classical theory, many different concepts have been introduced over time. They include Ramsey functions on graphs, many kinds of mixed Ramsey numbers, size Ramsey numbers, connected Ramsey numbers, anti-Ramsey numbers and Gallai-Ramsey numbers. These topics (and many others) can be found within the extensive mathematical literature. For an overview of classical graph Ramsey theory, the general surveys of Burr [1, 2], Radziszowski [19], Read and Wilson [21], and Sudakov [26] are invaluable. New directions and additional open questions in graph Ramsey theory are addressed in [4, 29, 30].

3. Calculating R(G, H) using propositional logic

For a mathematical introduction to logic, the reader is directed to [11]. We begin by recalling two concepts from propositional logic. A conjunctive normal form (CNF) is a Boolean expression consisting of a conjunction of disjunctions of propositional statements or their negations. An example of a CNF would be

\[(A \vee \neg B \vee \neg C) \wedge (\neg A \vee \neg B) \wedge (\neg A \vee \neg B \vee C),\]

where A, B and C represent propositional statements and ∨, ∧ and ¬ represent "OR," "AND" and "NOT," respectively. A Boolean expression is satisfiable if there is an assignment of "true" and "false" values to its propositional statements which makes the expression "true" (when evaluated using the standard truth table rules).

Cowen [9] used Mathematica's Boolean computational abilities to investigate some questions from graph Ramsey theory. In particular, he converted the problem of calculating R(Ks, Kt) into a satisfiabilty problem. Here is an overview of his approach.

Cowen's approach: Let s, t, and n be integers with 2 < s, t < n, and Kⁿ be the complete graph with vertices numbered 1, 2, . . . , n. For each pair of integers (i, j), where i, j ∈ V (Kn) with 1 ≤ i < j ≤ n, construct a CNF f (called "RamseyTest"), using variables ri,j and bi,j which denote the edge ei,j ∈ E(Kn) being colored red or blue, respectively. The CNF f consists of two sets of clauses:

• (Coloring clauses). For each pair of integers (i, j) with 1 ≤ i < j ≤ n, the clauses ri,j ∨ bi,j and ¬ri,j ∨ ¬bi,j provide a valid 2-coloring of Kn.
• (Non-monochromatic subgraph clauses). These clauses make sure that a 2-coloring of Kⁿ does not contain a monochromatic K^s or K^t .

If f is not satisfied, then Kⁿ → (Ks, Kt). In this case, R(Ks, Kt) ≤ n. If f is satisfied, then Kⁿ ↛ (Ks, Kt). In this case, n + 1 ≤ R(Ks, Kt).

Example 1. Suppose we wish to determine if R(K3, K3) ≤ 4. Using the Wolfram Language in Mathematica [28], the (combined) coloring and non-monochromatic subgraph clauses are, respectively:

Here, | | denotes "OR," && denotes "AND," and ! denotes "NOT." The two sets of clauses are connected by && in f. ♢

Cowen then improved the efficiency of his "RamseyTest" CNF by noting the following fact: In any 2-coloring of Kn, at least half of the edges from a particular vertex are of the same color (say, red). Thus, he created a new CNF f⁰ called "QuickerRamseyTest" which utilized this observation. This modest optimization decreased the runtime of "RamseyTest" significantly. For example, f⁰ determined that R(K4, K4) = 18 in approximately 4.5 seconds. In stark contrast to this, "RamseyTest" could not calculate R(K4, K4) in a reasonable amount of time and had to be manually

Out[5]= (red[{1, 2}] || blue[{1, 2}]) &&
       (! red[{1, 2}] || ! blue[{1, 2}]) &&
       (red[{1, 3}] || blue[{1, 3}]) &&
       (! red[{1, 3}] || ! blue[{1, 3}]) &&
       (red[{1, 4}] || blue[{1, 4}]) &&
       (! red[{1, 4}] || ! blue[{1, 4}]) &&
       (red[{2, 3}] || blue[{2, 3}]) &&
       (! red[{2, 3}] || ! blue[{2, 3}]) &&
       (red[{2, 4}] || blue[{2, 4}]) &&
       (! red[{2, 4}] || ! blue[{2, 4}]) &&
       (red[{3, 4}] || blue[{3, 4}]) && (! red[{3, 4}] || ! blue[{3, 4}])
Out[6]= (! red[{1, 2}] || ! red[{1, 3}] || ! red[{2, 3}]) &&
       (! red[{1, 2}] || ! red[{1, 4}] || ! red[{2, 4}]) &&
       (! red[{1, 3}] || ! red[{1, 4}] || ! red[{3, 4}]) &&
       (! red[{2, 3}] || ! red[{2, 4}] || ! red[{3, 4}]) &&
       (! blue[{1, 2}] || ! blue[{1, 3}] || ! blue[{2, 3}]) &&
       (! blue[{1, 2}] || ! blue[{1, 4}] || ! blue[{2, 4}]) &&
       (! blue[{1, 3}] || ! blue[{1, 4}] || ! blue[{3, 4}]) &&
       (! blue[{2, 3}] || ! blue[{2, 4}] || ! blue[{3, 4}])

terminated.

In [9], Cowen explored R(G1, G2, G3), where Gⁱ were complete graphs. We ask the natural question "Is it feasible to adapt Cowen's approach to compute new diagonal graph Ramsey numbers R(G, G), where G is any simple connected graph?"

Our hybrid approach: To further improve the efficiency of Cowen's "QuickerRamseyTest," we make additional use of Kn's symmetry when creating the 2-colorings of it. Let G be a simple connected graph. We want to decide if Kⁿ → (G, G) or not. Let k ≤ n. Now, fix k and consider G(k), the set of nonisomorphic simple graphs of order k. For a graph H ∈ G(k), color the edges of H in red and the edges of H (the complement of H) in blue. Then, embed H and H (with vertices v1, v2, . . . , vk) in Kⁿ (with vertices v1, v2, . . . , vn).

As in Cowen's approach, create a CNF f(H) using coloring clauses and non-monochromatic subgraph clauses. If f(H) is not satisfied for all H ∈ G(k), then Kⁿ → (G, G). In this case, R(G, G) ≤ n. If f(H) is satisfied for some H ∈ G(k), then Kⁿ ↛ (G, G). In this case, n + 1 ≤ R(G, G).

Example 2. Let G be a simple connected graph where |V (G)| = 7, say. Suppose that we want to determine if R(G, G) ≤ 9. There is a one-to-one correspondence between the set G(8) of nonisomorphic simple graphs of order eight and the set C(K8) of nonequivalent 2-colorings of K8.

Consider a simple graph H ∈ G(8) with k = 8 vertices (see Figure 1), with H being the complement of H. Associated with H is a partial 2-coloring C^H of K9, where the edges of H and H are red and blue, respectively. A CNF f(H) is created using clauses for the 2-colorings of K⁹ containing CH, along with clauses for the non-monochromatic subgraphs (∼= G) of K9. Then, f(H) is tested for satisfiability. Since there are 12346 nonisomorphic simple graphs with eight vertices, 12346 CNFs f(Hi) in total need to be constructed and tested for satisfiability. If all of the f(Hi) are unsatisfied, then R(G, G) ≤ 9. If at least one of the f(Hi) is satisfied, then R(G, G) > 9. ♢

Figure 1: The graph on the left is an H ∈ G(8). The graph on the right is a partial 2-coloring of K9. There, the solid edges are red, the dashed edges are blue and the dotted edges are not predetermined.

From a computational point of view, one wishes to make the most of symmetry when creating the 2-colorings of Kn. Fix k ≤ n. Recall that G(n) and C(Kn) denote the set of nonisomorphic simple graphs of order n and the set of nonequivalent 2-colorings of Kn, respectively. As alluded to in Example 2, there is a bijection between G(n) and C(Kn). To construct only the nonequivalent 2-colorings of Kn, we would have to encode the set G(n) within our computational programs. Unfortunately, even for somewhat small values of n (≥ 9), this cannot be accomplished in a practical way (see Table 1), since |G(n)| is very large.

In our hybrid approach, we do not create all of the 2-colorings of Kⁿ nor just the nonequivalent ones. Instead, we take the middle ground and construct the set of 2-colorings of Kⁿ which contain c, for each c ∈ C(Kk). This improves upon Cowen's approach in different ways. First, the number of 2-colorings of Kⁿ which need to be checked (when calculating R(G, G)) is greatly reduced. Second, the number of propositional variables and clauses in our Kⁿ 2-coloring CNF is decreased.

After collecting computational runtime data, we decided to create two sets of CNFs, namely F⁷ and F8. In particular, for k = 7 and 8, each CNF in F^k is made up of a set of clauses for the subgraphs (∼= G) of Kn, along with a set of clauses for the 2-colorings of Kⁿ containing a distinct c ∈ C(Kk). As such, |F7| = 1044 and |F8| = 12346 (see Table 1).

G(7) and G(8) were chosen for two reasons:

• |G(9)| = 274668. Thus, 274668 CNFs would need to be tested for satisfiability. This would drastically increase the computational runtime.
• Suppose that G(4), G(5) or G(6) is used. Then, Cowen's f⁰ actually fixes more monochromatic edges in a 2-coloring of Kⁿ (for 15 ≤ n ≤ 30, approximately), compared to the CNFs in our hybrid approach. In this case, it is better to use Cowen's f⁰ to calculate R(G, G).

Testing all of the CNFs in F⁷ (or F8) for satisfiability results in a longer runtime than f0, on smaller Kn. However, it can be sped up by running the individual f(Hi) in parallel. Since parallelization typically requires an abundance of RAM, the full potential of our hybrid approach (using F⁷ or F8) can only be realized on a high performance computing platform (HPC). That said, even without parallelization, the use of F⁷ (or F8) improves on Cowen's f0.

The calculations in this paper were obtained using several independent computing platforms:

• 3.68 GHz AMD Ryzen-9 3950X, 128GB RAM
• 3.3 GHz Intel i7-5820K, 32GB RAM
• 1.60 GHz Quad-Core Intel (R), 8GB RAM

Example 3. We calculated R(G603, G603) (see Figure 2) using F8, as well as f0. Our hybrid approach determined that R(G603, G603) = 15 in approximately 32000 minutes, while f⁰ was manually terminated after 33800 minutes. Here, the functions in F⁸ were sequentially tested (not in parallel) for satisfiability. The reader should note that R(G603, G603) was previously unknown. ♢

From experimentation, f⁰ appears to have a maximum subgraph clause size of roughly 3 million. Under that threshold, f⁰ runs much faster than testing all the functions in F⁷ (or F8), even when parallelized. In calculating R(G, G), either f0, F⁷ or F⁸ is used, depending on the subgraph clause size for G.

Our constrained coloring approach: Suppose we want to decide if R(G, G) ≤ n or not. Let H be a graph with known R(H, H) = k, where k ≤ n. Then, any 2-coloring of Kⁿ must contain a monochromatic (say, red) H. This reduces the number of 2-colorings of Kⁿ which need to be examined, when calculating R(G, G). Additional constraints can be imposed by embedding other monochromatic graphs within any 2-coloring of Kⁿ and/or by mathematical arguments. We create a new CNF which uses a partial 2-coloring of Kⁿ (inputted by the user).

The constrained coloring approach is a natural one and can provide a surprising decrease in computational runtimes. In Example 3, we mentioned that R(G603, G603) = 15 was determined in 30808 minutes, using our hybrid approach. However, using our constrained coloring approach, we are able to calculate R(G603), G603) in approximately 5.4 minutes. The proofs of both Theorems 1 and 2 use the constrained coloring approach.

With the hybrid and constrained coloring approaches, we obtain new diagonal graph Ramsey numbers. These results are found in Section 4.

Table 1: The number of nonisomorphic simple graphs of order n.

4. New diagonal graph Ramsey numbers

The following known results give lower bounds for R(G, G). They are used in our calculations.

Theorem B. (Chvatal-Harary ´ [5]). Let G and H be two graphs with no isolated vertices. Then, R(G, H) ≥ (χ(G) − 1)(n(H) − 1) + 1, where χ(G) is the chromatic number of G and n(H) is the order of the largest connected component of H.

Theorem C. (Chvatal-Harary ´ [6]). R(G, G) > (s · 2 |E(G)|−1 ) 1/|V (G)| , where s is the number of automorphisms of G.

Theorem D. (Burr-Erdos˝ [3]). Let |V (G)| ≥ 4. Then, R(G, G) ≥ ⌊(4 · |V (G)| − 1)/3⌋ for any connected G, and R(G, G) ≥ 2 · |V (G)| − 1 for any connected non-bipartite G.

In addition to [21], the following summary of known diagonal graph Ramsey numbers is given on page 62 of [19]:

• R(G, G), for all G without isolates on at most 4 vertices.
• R(G, G), for all G without isolates and with at most 7 edges.
• R(G, G), for all G on 5 vertices and with 7 or 8 edges.

Since we wish to calculate new diagonal graph Ramsey numbers, our attention is focused on connected simple graphs G, where |V (G)| ≥ 6 and |E(G)| ≥ 8.

In [21], it was conjectured that R(G603, G603) = 15 = R(G606, G606). See Figure 2. We prove this in Theorems 1 and 2.

Figure 2: Graphs G603 and G606, respectively.

Theorem 1. R(G603, G603) = 15.

Proof. In [21], it is stated that the diagonal Ramsey number of graph G603 − {v7} is 15. So, 15 ≤ R(G603, G603).

We wish to show that R(G603, G603) ≤ 15. Let v1, v2, . . . , v¹⁵ be an ordering of the vertices of K15. Assume that K¹⁵ ↛ (G603, G603). Then, there exists a 2-coloring C of K¹⁵ where every G603 subgraph is not monochromatic. In particular, there exists a G603 subgraph H which has a monochromatic G603 − {v7} and edge v5v⁷ of opposite color under C. Otherwise, we reach a desired contradiction. Without loss of generality, H has vertices and edges as described in Figure 2, the G603−{v7} in H is red and edge v5v⁷ is blue. This implies that edges v5vk, for 8 ≤ k ≤ 15, are blue. Otherwise, there would be a red G603 subgraph under C; thus giving a desired contradiction.

Let K be the graph G128, as found in [21]. There, it is stated that R(K, K) = 8. Now, consider the vertices v7, v8, v9, v10, v11, v12, v¹³ and v14. Thus, there is a red subgraph K, say {v7v8, v8v9, v9v10, v10v7, v10v11, v11v12, v12v7}, under C. Otherwise, there would be a blue G603 subgraph under C; thus giving a desired contradiction.

Next, consider the edges v12v1, v12v2, v12v³ and v12v4. If all four of these edges are red, then there would be a red G603 subgraph under C; thus giving a desired contradiction. Without loss of generality, the edge v12v¹ is blue. Figure 3 illustrates the partial 2-coloring C of K15.

Finally, several independent computer programs (using the constrained coloring approach described in Section 3) were written to check for a monochromatic subgraph G603 in all possible 2-colorings of K15, under the edge coloring constraints of C. In all instances, we obtain a monochromatic subgraph G603; thus giving a desired contradiction.

Hence, our assumption that K¹⁵ ↛ (G603, G603) was wrong. This, along with the fact that 15 ≤ R(G603, G603), implies that R(G603, G603) = 15.

Theorem 2. R(G606, G606) = 15.

Proof. In [21], it is stated that the diagonal Ramsey number of graph G606 − {v7} is 15. So, 15 ≤ R(G606, G606).

Figure 3: This is the partial 2-coloring C of K¹⁵ described in the proof of Theorem 1. The solid edges are red and the dashed edges are blue.

We wish to show that R(G606, G606) ≤ 15. Let v1, v2, . . . , v¹⁵ be an ordering of the vertices of K15. Assume that K¹⁵ ↛ (G606, G606). Then, there exists a 2-coloring C of K¹⁵ where every G606 subgraph is not monochromatic. In particular, there exists a G606 subgraph H which has a monochromatic G606 − {v7} and edge v4v⁷ of opposite color under C. Otherwise, we reach a desired contradiction. Without loss of generality, H has vertices and edges as described in Figure 2, the G606−{v7} in H is red and edge v4v⁷ is blue. This implies that edges v4vk, for 8 ≤ k ≤ 15, are blue. Otherwise, there would be a red G606 subgraph under C; thus giving a desired contradiction.

Let K be the graph G325, as found in [21]. There, it is stated that R(K, K) = 9. Now, consider the vertices v7, v8, v9, v10, v11, v12, v13, v¹⁴ and v15. Thus, there is a red subgraph K, say {v8v9, v9v10, v10v11, v11v8, v9v7, v11v12, v11v13}, under C. Otherwise, there would be a blue G606 subgraph under C; thus giving a desired contradiction.

Next, consider the edges v15v³ and v14v3. If both of these edges are red, then there would be a red G606 subgraph under C; thus giving a desired contradiction. Without loss of generality, the edge v15v³ is blue. Now, consider the edges v13v¹ and v12v1. If both of these edges are red, then there would be a red G606 subgraph under C; thus giving a desired contradiction. Without loss of generality, the edge v13v¹ is blue. Lastly, consider the edges v10v² and v8v2. If both of these edges are red, then there would be a red G606 subgraph under C; thus giving a desired contradiction. Without loss of generality, the edge v10v² is blue. Figure 4 illustrates the partial 2-coloring C of K15.

Finally, several independent computer programs (using the constrained coloring approach described in Section 3) were written to check for a monochromatic subgraph G606 in all possible 2-colorings of K15, under the edge coloring constraints of C. In all instances, we obtain a monochromatic subgraph G606; thus giving a desired contradiction.

Hence, our assumption that K¹⁵ ↛ (G606, G606) was wrong. This, along with the fact that 15 ≤ R(G606, G606), implies that R(G606, G606) = 15.

Figure 4: This is the partial 2-coloring C of K¹⁵ described in the proof of Theorem 2. The solid edges are red and the dashed edges are blue.

Table 2 gives additional diagonal graph Ramsey numbers, which were previously unknown when [21] was published, and which are not found in the current mathematical literature. These new results were obtained using the computational methods described in Section 3 of this paper.

Table 2: Some new diagonal graph Ramsey numbers. The "Gxxx" labels in this table correspond to classification numbers used in [21].

5. Miscellany

While exploring various graph Ramsey theory problems with Mathematica, Cowen [8] proved the following theorem. This beautiful result is what originally motivated our research project.

Theorem E. (Cowen [8]). Suppose R(Ks, Kt) = n. Then, G = Kⁿ − {uv} has a red/blue edge coloring where there is neither a red K^s nor a blue K^t .

To conclude this paper, we extend Theorem E to r colors.

Theorem 3. Suppose R(K^x1 , K^x2 , . . . , K^xk ) = n. Then, G = Kⁿ − {uv} has an r-coloring of the edges where no monochromatic K^xi exist in G.

Proof. Let G = Kⁿ − {uv}. Since G − {u} is Kn−1, it has an r-coloring of the edges where no monochromatic K^xi exist in G. To complete the coloring of G, we color an edge ux, x ∈ V \{u, v} using the color of edge vx. We now claim that no monochromatic K^xi exist in G. Suppose (to the contrary) a monochromatic K^xi exists in G. Then, K^xi does not contain both u and v. Without loss of generality, u /∈ K^xi . Then, K^xi is a monochromatic subgraph of G − {u}. This gives a desired contradiction and establishes the claim.

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