Ramsey Numbers on Posets (of Boolean Algebras)
by
Greg McColm
University of South Florida

It is known that for any finite weak order H, and any posets P, Q, there exists a poset R such that any 2-coloring of the H-isomorphic subposets of R admits a P-subposet of R whose H-subposets are all RED or a Q-subposet of R whose H-subposets are all BLUE. (Write R --> (P, Q)^H. If H is the poset of one node, write R --> (P, Q).) It is also known that if S and T are trees of rank <= \omega, and × is Cartesian product, then S ×T --> (S, T).

In this talk, we find that there exist finite posets P, Q such that P ×Q \not --> (P, Q). If B_m is the poset of P[m], then for any \alpha > 0, there exists n such that B_n+\alpha \not --> (B_n, B_\alpha). This contrasts with B_{\alphan + \alpha+ n} --> (B_n, B_\alpha) for all \alpha, n.

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Date received: March 26, 2001