Triangle Free Sets and Arithmetic Progressions - Two Pisier Type Problems

Let Pf (N ) be the set of finite nonempty subsets of N and for F,G ∈ Pf (N ) write F < G when maxF < minG. Let X = {(F,G) : F,G ∈ Pf (N ) and F < G}. A triangle in X is a set of the form {(F ∪ H,G), (F,G), (F,H ∪ G)} where F < H < G. Motivated by a question of Erdős, Nešetŕıl, and Rödl regarding three term arithmetic progressions, we show that any finite subset Y of X contains a relatively large triangle free subset. Exact values are obtained for the largest triangle free sets which can be guaranteed to exist in any set Y ⊆ X with n elements for all n ≤ 14. * This author acknowledges support received from the National Science Foundation (USA) via grant DMS-0070593. the electronic journal of combinatorics 9 (2002), #R22 1