New Lower Bounds for Some Multicolored Ramsey Numbers

We use finite fields and extend a result of Fan Chung to give eight new, nontrivial, lower bounds.Comment: 6 page

Lower bounds for some Ramsey numbers

AbstractLet n, r, u1, u2,…,uk be positive integers satisfying ui ⩾ r for i = 1,2,…,k. The symbol n → (u1, u2,…,uk)r means that, for any partition of the r-subsets of an n-set S into k classes C1, C2,…,Ck, there is a ui-subset of S all of whose r-subsets belong to Ci for some i, 1 ⩽ i ⩽ k. A theorem of F.P. Ramsey asserts that, if r, u1, u2,…,uk are given, then n → (u1, u2,…,uk)r for all sufficiently large n. n ↦ (u1, u2,…,uk)r denotes the negation of n → (u1, u2,…,uk)r. In this paper a number of results of the form n ↦ (u1, u2,…,uk)3 are obtained

Difference Ramsey Numbers and Issai Numbers

We present a recursive algorithm for finding good lower bounds for the classical Ramsey numbers. Using notions from this algorithm we then give some results for generalized Schur numbers, which we call Issai numbers.Comment: 10 page

Constructive Lower Bounds on Classical Multicolor Ramsey Numbers

This paper studies lower bounds for classical multicolor Ramsey numbers, first by giving a short overview of past results, and then by presenting several general constructions establishing new lower bounds for many diagonal and off-diagonal multicolor Ramsey numbers. In particular, we improve several lower bounds for R_k(4) and R_k(5) for some small k, including 415 \u3c = R_3(5), 634 \u3c = R_4(4), 2721 \u3c = R_4(5), 3416 \u3c = R_5(4) and 26082 \u3c = R_5(5). Most of the new lower bounds are consequences of general constructions

Applying a genetic algorithm to improve the lower bounds of multi-color ramsey numbers

Ramsey Theory studies conditions when a combinatorial object contains necessarily some smaller given objects. The role of Ramsey Numbers is to quantify some of the general existential theorems in Ramsey Theory. The objective of this thesis is to try to improve the lower bounds of Ramsey Numbers, in particular the bounds for multi-color graph Ramsey Numbers. Let Gi,G2,---,Gm be graphs on some n. R(G\,G2, ,Gm) denotes the m-color Ramsey number for graphs, avoiding G, in color i for 1 \u3c i \u3c m. Thus, to show R(G\, G2, . , Gm) \u3e N, we need to find an edge coloring for a Kn graph using m colors (for N as large as possible) avoiding G, in color i. An order-based Genetic Algorithm (GA) combined with a greedy coloring heuristic is used as a search heuristic in finding the coloring. Each chromo some is a permutation representing the order in which the edges of graph are colored. (This approach was far more successful than a string representing the coloring.) The algorithm was successful. The known lower bounds for R(C4,C4,C4), R(C4,C4, K3), R(C4,K3, K3), R(C5,C5, C5) were matched, and two new lower bounds for Ramsey numbers, R(C4,C4,K3,K3) \u3e 25 and R(Cs, C4, K3) \u3e 13, were found. *See thesis for correct equations and mathematical term

Some properties of Ramsey numbers

AbstractIn this paper, some properties of Ramsey numbers are studied, and the following results are presented. 1.(1) For any positive integers k1, k2, …, km l1, l2, …, lm (m > 1), we have r ∏i=1m ki + 1, ∏i=1m li + 1 ≥ ∏i=1m [ r (ki + 1,li + 1) − 1] + 1.2.(2) For any positive integers k1, k2, …, km, l1, l2, …, ln , we have r ∑i=1m ki + 1, ∑j=1n lj + 1 ≥ ∑i=1m∑j=1n r (ki + 1,lj + 1) − mn + 1. Based on the known results of Ramsey numbers, some results of upper bounds and lower bounds of Ramsey numbers can be directly derived by those properties