On the minimum degree of minimal Ramsey graphs for multiple colours

A graph G is r-Ramsey for a graph H, denoted by G\rightarrow (H)_r, if every r-colouring of the edges of G contains a monochromatic copy of H. The graph G is called r-Ramsey-minimal for H if it is r-Ramsey for H but no proper subgraph of G possesses this property. Let s_r(H) denote the smallest minimum degree of G over all graphs G that are r-Ramsey-minimal for H. The study of the parameter s_2 was initiated by Burr, Erd\H{o}s, and Lov\'{a}sz in 1976 when they showed that for the clique s_2(K_k)=(k-1)^2. In this paper, we study the dependency of s_r(K_k) on r and show that, under the condition that k is constant, s_r(K_k) = r^2 polylog r. We also give an upper bound on s_r(K_k) which is polynomial in both r and k, and we determine s_r(K_3) up to a factor of log r

On a Diagonal Conjecturefor Classical Ramsey Numbers

Let R(k1, · · · , kr) denote the classical r-color Ramsey number for integers ki ≥ 2. The Diagonal Conjecture (DC) for classical Ramsey numbers poses that if k1, · · · , kr are integers no smaller than 3 and kr−1 ≤ kr, then R(k1, · · · , kr−2, kr−1 − 1, kr + 1) ≤ R(k1, · · · , kr). We obtain some implications of this conjecture, present evidence for its validity, and discuss related problems. Let Rr(k) stand for the r-color Ramsey number R(k, · · · , k). It is known that limr→∞ Rr(3)1/r exists, either finite or infinite, the latter conjectured by Erd˝os. This limit is related to the Shannon capacity of complements of K3-free graphs. We prove that if DC holds, and limr→∞ Rr(3)1/r is finite, then limr→∞ Rr(k) 1/r is finite for every integer k ≥ 3

The Wonder of Colors and the Principle of Ariadne

The Principle of Ariadne, formulated in 1988 ago by Walter Carnielli and Carlos Di Prisco and later published in 1993, is an infinitary principle that is independent of the Axiom of Choice in ZF, although it can be consistently added to the remaining ZF axioms. The present paper surveys, and motivates, the foundational importance of the Principle of Ariadne and proposes the Ariadne Game, showing that the Principle of Ariadne, corresponds precisely to a winning strategy for the Ariadne Game. Some relations to other alternative. set-theoretical principles are also briefly discussed