An Anti-Ramsey Theorem on Posets

Abstract

It is known that if P and Q are posets and * is lexicographic product, then (in t he Erdos-Rado partition notation) , P*Q ➔ (P, Q) . It is known that if Sand Tare trees of r ank at most w, and " x " is Cartesian product, then S x T ➔ (S, T). In this article we exhibit pairs of finite posets P and Q such that P x Q I+ (P, Q). In particular, we prove t hat if Bn is the poset of the power set on n elements, t hen for each integer a, > 1, t here exists N such t hat n > N implies B n+o f+ (Bn , B0 ) ; indeed, we can choose n such that B n+o f+ (En, B2). We conclude by looking at a few positive results.