New Lower Bounds for Some Multicolored Ramsey Numbers

Aaron Robertson^{1} Department of Mathematics, Temple University Philadelphia, PA 19122 email: aaron@math.temple.edu

Submitted: October 26, 1998; Accepted November 15, 1998 Classification: 05D10, 05D05

Abstract

In this article we use two different methods to find new lower bounds for some multicolored Ramsey numbers. In the first part we use the finite field method used by Greenwood and Gleason [GG] to show that R(5, 5, 5) ≥ 242 and R(6, 6, 6) ≥ 692. In the second part we extend Fan Chung’s result in [C] to show that,

R(3, 3, 3, k_{1}, k_{2}, …, k_{r}) ≥ 3R(3, 3, k_{1}, k_{2}, …, k_{r}) + R(k_{1}, k_{2}, …, k_{r}) − 3

holds for any natural number r and for any k_{i} ≥ 3, i = 1, 2, … r. This general result, along with known results, imply the following nontrivial bounds: R(3, 3, 3, 4) ≥ 91, R(3, 3, 3, 5) ≥ 137, R(3, 3, 3, 6) ≥ 165, R(3, 3, 3, 7) ≥ 220, R(3, 3, 3, 9) ≥ 336, and R(3, 3, 3, 11) ≥ 422.

Introduction

This paper is presented in two part which can be read independently of each other. Part one uses finite fields, and Part two extends an argument by Fan Chung.

Part one of this article is accompanied by the Maple package RES, available for download at the author’s website.

Recall that N = R(k_{1}, k_{2}, …, k_{r}) is the minimal integer with the following property:

Ramsey Property: If we r-color the edges of the complete graph on N vertices, then there exists j, 1 ≤ j ≤ r, such that a monochromatic j-colored complete graph on k_{j} vertices is a subgraph of the r-colored K_{N}.

1webpage: www.math.temple.edu/˜ aaron This paper is part of the author’s Ph.D. thesis under the direction of Doron Zeilberger.

This paper was supported in part by the NSF under the PI-ship of Doron Zeilberger.

Part One: The Finite Field Method

In this first part we add two more lower bounds to Radziszowski’s Dynamic Survey [R] on the subject. We show, by using the finite field technique in [GG], that R(5, 5, 5) ≥ 242 and R(6, 6, 6) ≥ 692. The previous best lower bound for R(5, 5, 5) was 169 given by Song [S], who more generally shows that R(\underbrace{5, 5, …, 5}_{r times}) ≥ 4(6.48)^{r−1} + 1

holds for all r. For R(6, 6, 6) there was no established nontrivial lower bound.

Consider the number R(5, 5, 5). To find a lower bound, L, we are searching for a three coloring of K_{L} which avoids a monochromatic K_{5}. We use an argument of Greenwood and Gleason, which is reproduced here for the sake of completeness.

Let L be prime and consider the field of L elements, numbered from 0 to L − 1. Associate each field element with a vertex of K_{L}. We require that 3 divides L − 1. Now consider the cubic residues of the multiplicative group Z_{L}^{∗} = Z_{L} \ {0}, which form a coset of Z_{L}^{∗}. Note that since 3 divides L − 1, there must be 2 other cosets.

Let i and j be two vertices of K_{L}. Color the edges of K_{L} as follows: If j − i is a cubic residue color the edge connecting i and j red, if it is in the second coset, color the edge blue, and if it is the third coset, color the edge green. (Note that the order of differencing is immaterial since −1 is a cubic residue.)

Now suppose that a monochromatic K_{5} exists in this coloring. Without loss of generality we may call the five vertices 0, a, b, c, and d, with 0 < a < b < c < d. Then the set of edges, E = {a, b, c, d, b − a, c − a, d − a, c − b, d − b, d − c}, must be a subset of one of the cosets. Since a ≠ 0, multiplication by a^{−1} is allowed. Set B = ba^{−1}, C = ca^{−1}, and D = da^{−1}. Then the set a^{−1}E = {1, B, C, D, B − 1, C − 1, D − 1, C − B, D − B, D − C} must be a subset of the cubic residues. Hence if we find an L for which there does not exist B, C, and D such that a^{−1}E is a subset of the cubic residues, then we can conclude that R(5, 5, 5) > L. Of course, this argument holds for R(\underbrace{t, t, …, t}_{k times}) for any k, and any t.

Using RES

We only acheived results when we restricted our search to fields of prime order (although any finite field can be explored using RES (or at least easily modified to do so)). Since we are considering the number R(5, 5, 5), reject any prime, q, for which 3 does not divide q − 1. This can be accomplished automaticly by using the procedure pryme. By using the procedure res we produce all of the cubic residues of Z_{p}^{∗}, for a given prime, p. We then use the procedure siv to discard any residue, R, for which R − 1 is not a residue. We now have a much more manageable list to search. We then call the procedure diffcheck to choose all possible 3-sets (for B, C, and D) and check whether or not the differences between any two elements are all cubic residues. If such a 3-set exists, diffcheck will output the first 3-set it finds. However, in the event that no such 3-set exists, diffcheck will output 1.

RES can also be used to search finite fields whose order is not prime. For example, to verify that the field on 2^{4} elements, avoids a monochromatic triangle by using cubic residues (this fact was proven in [GG]), type GalField3(2,4,3).

By using RES we were able to find the following lower bounds: R(5, 5, 5) ≥ 242 and R(6, 6, 6) ≥ 692. These are obtained by the following colorings: (Since −1 is a cubic residue it suffices to list only entries up to 120 for R(5, 5, 5) and 345 for R(6, 6, 6).)

R(5, 5, 5) > 241:

Color 1: 1, 5, 6, 8, 17, 21, 23, 25, 26, 27, 28, 30, 33, 36, 40, 41, 43, 44, 47, 48, 57, 61, 64, 73, 76, 79, 85, 87, 91, 93, 98, 101, 102, 103, 105, 106, 111, 115, 116, 117

Color 2: 2, 7, 9, 10, 11, 12, 16, 19, 29, 31, 34, 35, 37, 39, 42, 45, 46, 50, 52, 54, 55, 56, 59, 60, 66, 67, 71, 72, 80, 82, 83, 86, 88, 89, 94, 95, 96, 113, 114, 119

Color 3: 3, 4, 13, 14, 15, 18, 20, 22, 24, 32, 38, 49, 51, 53, 58, 62, 63, 65, 68, 69, 70, 74, 75, 77, 78, 81, 84, 90, 92, 97, 99, 100, 104, 107, 108, 109, 110, 112, 118, 120

R(6, 6, 6) > 691:

Color 1: 1, 2, 4, 5, 8, 10, 16, 19, 20, 21, 25, 27, 31, 32, 33, 38, 39, 40, 42, 50, 51, 54, 62, 64, 66, 67, 69, 71, 73, 76, 78, 80, 83, 84, 87, 89, 95, 100, 102, 105, 107, 108, 109, 123, 124, 125, 128, 132, 134, 135, 138, 139, 142, 146, 149, 151, 152, 155, 156, 160, 163, 165, 166, 168, 173, 174, 178, 179, 181, 190, 191, 195, 199, 200, 204, 210, 214, 216, 218, 246, 248, 250, 255, 256, 259, 263, 264, 268, 270, 271, 276, 278, 283, 284, 291, 292, 293, 298, 301, 302, 304, 309, 310, 311, 312, 320, 326, 329, 330, 332, 333, 335, 336, 343, 345

Color 2: 7, 9, 11, 13, 14, 17, 18, 22, 23, 26, 28, 29, 34, 35, 36, 41, 44, 45, 46, 52, 55, 56, 58, 65, 68, 70, 72, 82, 85, 88, 90, 92, 97, 103, 104, 110, 111, 112, 115, 116, 127, 129, 130, 131, 133, 136, 140, 141, 144, 145, 147, 159, 164, 167, 170, 171, 175, 176, 177, 180, 183, 184, 189, 194, 197, 205, 206, 208, 209, 217, 220, 222, 224, 225, 227, 229, 230, 231, 232, 233, 237, 241, 243, 247, 251, 254, 257, 258, 260, 262, 266, 272, 273, 275, 279, 280, 281, 282, 288, 290, 294, 297, 303, 313, 318, 323, 325, 328, 331, 334, 337, 339, 340, 341, 342

Color 3: 3, 6, 12, 15, 24, 30, 37, 43, 47, 48, 49, 53, 57, 59, 60, 61, 63, 74, 75, 77, 79, 81, 86, 91, 93, 94, 96, 98, 99, 101, 106, 113, 114, 117, 118, 119, 120, 121, 122, 126, 137, 143, 148, 150, 153, 154, 157, 158, 161, 162, 169, 172, 182, 185, 186, 187, 188, 192, 193, 196, 198, 201, 202, 203, 207, 211, 212, 213, 215, 219, 221, 223, 226, 228, 234, 235, 236, 238, 239, 240, 242, 244, 245, 249, 252, 253, 261, 265, 267, 269, 274, 277, 285, 286, 287, 289, 295, 296, 299, 300, 305, 306, 307, 308, 314, 315, 316, 317, 319, 321, 322, 324, 327, 338, 344

Part Two: On the Ramsey Numbers R(3, 3, 3, k_{1}, k_{2}, …, k_{r})

Let N = R(k_{1}, k_{2}, …, k_{r}). The Ramsey Property implies that there must exist a graph on N − 1 vertices which avoids the Ramsey Property. Using such a graph, along with the construction in [C], we will prove that, for any natural number r and for any k_{i} ≥ 3, i = 1, 2, … r,

R(3, 3, 3, k_{1}, k_{2}, …, k_{r}) ≥ 3R(3, 3, k_{1}, k_{2}, …, k_{r}) + R(k_{1}, k_{2}, …, k_{r}) − 3.

The Construction

Fix r ≥ 1, and k_{i} ≥ 3 for i = 1, 2, … r. Let M = R(3, 3, k_{1}, k_{2}, … k_{r}). Then there exists a graph, G, on M − 1 vertices which avoids the Ramsey Property. Call the incidence matrix of this graph T_{r+2} = T_{r+2}(x_{0}, x_{1}, x_{2}, …, x_{r+2}), where x_{0} are the diagonal entries only, and the x_{i}, for i = 1, 2, … r + 2, are the r + 2 colors. By definition of G, there are no x_{1}-colored nor x_{2}-colored triangles, and no x_{i+2}-colored K_{k_{i}}, for i = 1, 2, … r.

Now consider the following slightly modified construction from [C]:

A
D  B
T_{r+3}(0, 1, 2, …, r + 3) = E F C
1, …, 1 2, …, 2 3, …, 3
⋮  ⋮  ⋮  G
1, …, 1 2, …, 2 3, …, 3

the incidence matrix of a graph H on 3M + R(k_{1}, k_{2}, …, k_{r}) − 4 vertices, where

A = T_{r+2}(0, 2, 3, 4, 5, …, r + 3)
B = T_{r+2}(0, 3, 1, 4, 5, …, r + 3)
C = T_{r+2}(0, 1, 2, 4, 5, …, r + 3)
D = T_{r+2}(3, 2, 1, 4, 5, …, r + 3)
E = T_{r+2}(2, 1, 3, 4, 5, …, r + 3)
F = T_{r+2}(1, 3, 2, 4, 5, …, r + 3)

and G is any matrix on R(k_{1}, k_{2}, …, k_{r}) − 1 vertices in the colors 4 through r + 3 which avoids the Ramsey Property.

Using Fan Chung’s result [C] we see that the graph H avoids 1-colored, 2-colored, and 3-colored triangles. We now argue that no (j + 3)-colored K_{k_{j}} exists in H for j = 1, 2, …, r: Assume there exists a J-colored K_{k_{J}} in H, for some J between 4 and r +3. Then there must exist \binom{k_{J}}{2} entries in the 3(M −1)×3(M −1) upper left submatrix of T_{r+3}, all of value J, which form the edges of a complete graph on k_{J} vertices. However, by the construction of T_{r+3} we see that all of these entries can be taken modulo M −1, since the entries of value J in each block are in exactly the same places as in the upper left block, A. We further note that if (s, t) and (u, v) are two of the entries in question, then (s, t) ≢ (u, v) (mod M −1) (componentwise). Without loss of generality we may assume s < u. If we had s ≡ u (mod M −1), then since (u, s) must also have the same value as (u, v), we would have (u, s) ≡ (u, u) (mod M −1). This implies that the entry J is on the diagonal of T_{r+2}(0, 2, 3, 4, …, r + 3), a contradiction. Hence, if we have a J-colored K_{k_{J}} in the upper left submatrix of T_{r+3} (of size 3(M − 1) × 3(M − 1)), then there must be a J-colored K_{k_{J}} in T_{r+2}(0, 2, 3, 4, …, r + 3), contradicting the definition of T_{r+2}.

Remark: Up to the renaming of colors and vertices, the above permutation configuration of colors which defines T_{r+3} is the only configuration which will avoid monochromatic triangles.

Harvesting Some Lower Bounds for Ramsey Numbers

It is amazing that the result of this section has not been observed for the 25 years since [C] was published. Using this observation we will give 6 new lower bounds. Currently in Radziszowski’s Survey [R], we have that R(3, 3, 3, 4) ≥ 87, due to Exoo [E1]. By applying the result of this section to R(3, 3, 3, 4) and using the fact that R(3, 3, 4) ≥ 30 [K], we get the new lower bound: R(3, 3, 3, 4) ≥ 91. Using the bound R(3, 3, 5) ≥ 45 [E2,KLR], we get the bound R(3, 3, 3, 5) ≥ 137. Finally, using R(3, 3, 6) ≥ 54, R(3, 3, 7) ≥ 72, R(3, 3, 9) ≥ 110, and R(3, 3, 11) ≥ 138 all from [SLZL], we get the lower bounds R(3, 3, 3, 6) ≥ 165, R(3, 3, 3, 7) ≥ 220, R(3, 3, 3, 9) ≥ 336, and R(3, 3, 3, 11) ≥ 422.

Acknowledgments

I would like to thank my advisor, Doron Zeilberger, for his useful comments and insight regarding this paper. More importantly, however, I would like to thank him for sparking my interest in combinatorics and for sharing his mathematical philosophies. The mathematical community could benefit greatly from more mathematicians like him.

I would also like to thank Brendan McKay for his assistance and the referee for calling my attention to [SLZL].

References

[C] F. Chung, On the Ramsey Numbers N(3, 3, …, 3; 2), Discrete Mathematics, 5, 1973, 317-321.

[E1] G. Exoo, Some New Ramsey Colorings, Electronic Journal of Combinatorics, R29, 5, 1998, 5pp.

[E2] G. Exoo, Constructing Ramsey Graphs with a Computer, Congressus Numerantium, 59, 1987, 31-36.

[GG] R. Greenwood and A. Gleason, Combinatorial Relations and Chromatic Graphs, Canadian Journal of Mathematics, 7, 1955, 1-7.

[K] J. Kalbfleisch, Chromatic Graphs and Ramsey’s Theorem, (Ph.D. Thesis), University of Waterloo, January 1966.

[KLR] D.L. Kreher, Li Wei, and S. Radziszowski, Lower Bounds for Multi-Colored Ramsey Numbers from Group Orbits, Journal of Combinatorial Mathematics and Combinatorial Computing, 4, 1988, 87-95.

[R] S. Radziszowski, Small Ramsey Numbers, Electronic Journal of Combinatorics, Dynamic Survey DS1, 1994, 28pp.

[S] Song En Min, New Lower Bound Formulas for the Ramsey Numbers N(k, …, k; 2) (in Chinese), Mathematica Applicata, 6, 1993 suppl., 113-116.

[SLZL] Su Wenlong, Luo Haipeng, Zhang Zhengyou, and Li Guiqing, New Lower Bounds of Fifteen Classical Ramsey Numbers, to appear in Australasian Journal of Combinatorics.