On the Ramsey number R(3, 6)

The lower bound for the classical Ramsey number R(4, 6) is improved from 35 to 36. The author has found 37 new edge colorings of K35 that have no complete graphs of order 4 in the first color, and no complete graphs of order 6 in the second color. The most symmetric of the colorings has an automorphism group of order 4, with one fixed point, and is presented in detail. The colorings were found using a heuristic search procedure. This note deals with a new lower bound for the classical Ramsey number R(4, 6). Recall that the classical Ramsey number R(s, t) is the smallest integer n such that in any two-coloring of the edges of Kn there is a monochromatic copy of Ks in the first color or a monochromatic copy of Kt in the second color. A recent summary of the state of the art for Ramsey numbers can be found in the Dynamic Survey [10]. Theorem 1. R(4, 6) ≥ 36. Proof. The proof is given by the coloring of K35 which can be derived from Table 1, below. This table contains an adjacency list for the color one graph of a two-coloring of K35. All other edges are assigned color two. This coloring improves the lower bound for R(4, 6) from 35 to 36, five short of the upper bound of 41 [8]. The coloring has an automorphism group of order 4, with exactly one fixed point [7]. It was found using a method that the author had discarded many years ago, a method we now discuss. A number of the lower bounds given in the table of two color classical Ramsey numbers (see Table 1 in the Dynamic Survey [10]) were established by this author using computer search techniques. Three of these, R(4, 6), R(3, 10), and R(5, 5) [1, 2, 3], are the smallest unsettled cases for two color classical Ramsey numbers. They are also, in the opinion of this author, the only unsettled cases where optimal colorings can be found using computer the electronic journal of combinatorics 19 (2012), #P66 1 0: 2 6 7 9 11 13 15 17 18 20 21 23 24 26 28 32 1: 3 4 5 9 11 13 15 17 18 21 22 23 25 27 29 33 2: 0 4 5 8 10 12 14 16 19 20 21 22 25 27 28 32 3: 1 6 7 8 10 12 14 16 19 20 22 23 24 26 29 33 4: 1 2 7 8 10 11 13 17 19 20 22 24 26 31 34 5: 1 2 6 8 9 11 14 17 19 21 23 24 26 30 34 6: 0 3 5 9 10 11 12 16 18 21 23 25 27 31 34 7: 0 3 4 8 9 10 15 16 18 20 22 25 27 30 34 8: 2 3 4 5 7 9 12 15 17 23 24 27 29 31 32 9: 0 1 5 6 7 8 13 14 16 22 25 26 29 31 32 10: 2 3 4 6 7 11 13 14 18 21 25 26 28 30 33 11: 0 1 4 5 6 10 12 15 19 20 24 27 28 30 33 12: 2 3 6 8 11 13 15 17 18 20 22 25 31 33 13: 0 1 4 9 10 12 14 16 19 21 23 24 31 33 14: 2 3 5 9 10 13 15 17 18 20 22 24 30 32 15: 0 1 7 8 11 12 14 16 19 21 23 25 30 32 16: 2 3 6 7 9 13 15 19 21 24 27 29 32 34 17: 0 1 4 5 8 12 14 18 20 25 26 29 32 34 18: 0 1 6 7 10 12 14 17 22 24 27 28 33 34 19: 2 3 4 5 11 13 15 16 23 25 26 28 33 34 20: 0 2 3 4 7 11 12 14 17 23 27 29 30 31 21: 0 1 2 5 6 10 13 15 16 22 26 29 30 31 22: 1 2 3 4 7 9 12 14 18 21 26 28 30 31 23: 0 1 3 5 6 8 13 15 19 20 27 28 30 31 24: 0 3 4 5 8 11 13 14 16 18 26 27 32 33 34 25: 1 2 6 7 9 10 12 15 17 19 26 27 32 33 34 26: 0 3 4 5 9 10 17 19 21 22 24 25 28 29 27: 1 2 6 7 8 11 16 18 20 23 24 25 28 29 28: 0 2 10 11 18 19 22 23 26 27 29 31 34 29: 1 3 8 9 16 17 20 21 26 27 28 30 34 30: 5 7 10 11 14 15 20 21 22 23 29 34 31: 4 6 8 9 12 13 20 21 22 23 28 34 32: 0 2 8 9 14 15 16 17 24 25 33 33: 1 3 10 11 12 13 18 19 24 25 32 34: 4 5 6 7 16 17 18 19 24 25 28 29 30 31 Table 1: A (4, 6)-coloring of K35. the electronic journal of combinatorics 19 (2012), #P66 2 methods that manipilate colorings one edge at a time. These three lower bounds were established using one (or more) of the following computer search methods. In each case, the object of the method is to produce colorings with no monochromatic subgraphs of order s in color one, and no monochomatic subgraphs of order t in color two. Monochromatic copies of such subgraphs in a coloring will be referred to as bad subgraphs. Method A: One begins with a randomly generated edge coloring for a complete graph whose order n is small enough so that a good coloring can be obtained easily. Then using simulated annealing [6], or a synthesis of simulated annealing and tabu search [5], the coloring is transformed into a good coloring by looking at individual edges, and choosing the color that minimizes the number of bad subgraphs. When all bad subgraphs have been eliminated, n is incremented, and the process is repeated. Method B: This method is different from Method A in two respects. Instead of beginning with a graph of small order, one begins with a complete graph of the desired order (e.g., large enough to improve the lower bound for the Ramsey number). But the important difference pertains to the objective function. Instead of simply recoloring edges so as to minimize the number of bad subgraphs, we add a term to the objective that attempts to maximize the number of monochromatic induced copies of P4, the path on 4 vertices. The importance of the P4 count in the objective can be as great or greater than the bad subgraph counts. Method C: Here one begins by searching for highly symmetric colorings, for example circle colorings (or more generally, Cayley colorings), that have relatively few bad subgraphs, and that have one additional property. They must have individual edges (as opposed to orbits) that, when recolored, reduce the number of monochromatic subgraphs. Once such a coloring is found, one proceeds as in Method A. One further remark on Method B should be made. Often we used a somewhat more detailed variation of the method. There are 11 isomorphism classes of graphs of order 4, and hence 11 essentially different ways to two-color the edges of a subgraph of order 4 in a two-coloring of Kn. If we count the number of vertex sets of size four which induce each of these 11 possible colorings, we produce a vector of length 11. Edges can be thus recolored so as to minimize the distance between the computed vector and a postulated target vector. The target vector might be determined by looking at known good colorings, by a higher level optimization process, or by sheer speculation. Experience has shown that maximizing the number of induced P4’s is the key ingredient. All three methods have successfully produced lower bounds for classical Ramsey numbers. The current lower bound for R(5, 5), for example, was originally established [1] using Method C. Very soon thereafter, we were able to find the same coloring using Method B, and eventually, using Method A. Since the idea behind Method B does not seem to have any direct theoretical justification, and since Method A was faster (counting induced P4’s takes more time than counting cliques, when the number of cliques is small), Method B was gradually discarded. the electronic journal of combinatorics 19 (2012), #P66 3 4-subgraph G1 G2 G3 G4 G5 E4 0 0 0 0 0 K2 1886 1464 1475 1484 1500 2K2 164