The Erdos-Szekeres theorem tells us that every list of n^2+1 distinct numbers must contain either an increasing sequence of length n+1 or a decreasing sequence of length n+1. This suggests a two-player game (call the players I and D) whose state after k moves is a list of k distinct numbers, where the players take turns generating a new number via some random mechanism and inserting it somewhere in the list, such that player I (resp. player D) wins when the sequence contains an increasing (resp. decreasing) subsequence of length n+1. (I was thinking of giving the players actual names, but all I could come up with is Ivana and Donald, and I think all of us can use a break from politics.) The E-S theorem guarantees that someone will win within n^2+1 moves. One could play this game with a bag of tiles (Scrabble tiles would be a nice choice, since the number of letters in the alphabet is conveniently of the form n^2+1). Or one could roll a die repeatedly, sampling with replacement rather than without, and bound the length of the game using a variant of the Erdos-Szekeres theorem in which repeats are allowed and the subsequences are permitted to be weakly increasing/decreasing. Has anyone studied such a game? I just did some searching and found other Ramsey theory games but not ones involving an element of chance. Jim Propp