Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs

Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs

Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs

New Upper Bounds For A Canonical Ramsey Problem

. Let f(l; k) be the minimum n with the property that every coloring c : \Gamma [n+1] 2 \Delta ! f1; 2; : : : g yields either x0 ! \Delta \Delta \Delta ! x l with c(xo ; x1 ) = \Delta \Delta \Delta = c(x l\Gamma1 ; x l ), or y0 ! \Delta \Delta \Delta ! y k with c(y0 ; y1 ); : : : ; c(y k\Gamma1 ; y k ) all distinct. We prove that if k = o( p l ), then f(l; k) ¸ l k\Gamma1 as l ! 1. This supports the conjecture of Lefmann, Rodl, and Thomas that f(l; k) = l k\Gamma1 . 1. Canonical Colorings Ramsey Theory studies the monochromatic subgraphs that are forced to appear in every k-coloring of the edges of K n . If we relax the requirement on the number of available colors, we can still ask what types of subgraphs are forced. Erdos and Rado [1] formulated this question more precisely. Let N be the set of positive integers. If S ` N and k ? 0, then let \Gamma S k \Delta be the family of all k-subsets of S. Theorem 1 (Erdos-Rado [1]). Suppose that \Gamma N 2 \Delta is arbit..