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A (k, g)-cage is a k-regular graph of girth g of minimum order. In this survey, we present the results of over 50 years of searches for cages. We present the important theorems, list all the known cages, compile tables of current record holders, and describe in some detail most of the relevant constructions.
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
For k ≥ 5, we establish new lower bounds on the Schur numbers S(k) and on the k-color Ramsey numbers of K3. For integers m and n, let [m, n] denote the set {i | m ≤ i ≤ n}. A set S of integers is called sum-free if i, j ∈ S implies i + j ∈ S, where we allow i = j. The Schur function S(k) is defined for all positive integers as the maximum n such that [1, n](More)