On an Erdős-Szekeres Game

We consider a 2-player permutation game inspired by the celebrated Erdős-Szekeres Theorem. The game depends on two positive integer parameters $a$ and $b$ and we determine the winner and give a winning strategy when $a \geq b$ and $b \in \left{2,3,4,5\right}$.

💡 Research Summary

The paper studies a two‑player permutation game that is a misère version of the classic Erdős‑Szekeres theorem. In the original theorem, any sequence of length at least ((a-1)(b-1)+1) contains either an increasing subsequence of length (a) (denoted (I_a)) or a decreasing subsequence of length (b) (denoted (J_b)). The game, first introduced by Harary, Sagan and West (1983), proceeds as follows: starting from an empty permutation, the players alternately append a new integer from ({1,\dots ,n+1}) to the current permutation of length (n). The first player who creates an (I_a) or a (J_b) loses (misère rule).

The author introduces a visual representation inspired by Seidenberg’s proof. For each newly placed element (\pi_n) the pair ((c,r)) is recorded, where (c) is the length of the longest increasing subsequence ending at (\pi_n) and (r) is the length of the longest decreasing subsequence ending at (\pi_n). These pairs are plotted on a grid of size ((a-1)\times(b-1)). When a cell ((c,r)) is shaded, all cells ((c’,r’)) with (c’\le c) and (r’\le r) become “eliminated’’ and can never be used again. Proposition 1 shows that any legal next move must shade a cell that is edge‑adjacent to the current eliminated region, and conversely every such edge‑adjacent open cell corresponds to a possible move. This creates a contiguous eliminated region whose boundary can be read as a ternary word (the representation used in Albert et al.