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In 1935, ErdH{o}s and Szekeres proved that $(m-1)(k-1)+1$ is the minimum number of points in the plane which definitely contain an increasing subset of $m$ points or a decreasing subset of $k$ points (as ordered by their $x$-coordinates). We consider their result from an on-line game perspective: Let points be determined one by one by player A first determining the $x$-coordinate and then player B determining the $y$-coordinate. What is the minimum number of points such that player A can force an increasing subset of $m$ points or a decreasing subset of $k$ points? We introduce this as the ErdH{o}s-Szekeres on-line number and denote it by $text{ESO}(m,k)$. We observe that $text{ESO}(m,k) < (m-1)(k-1)+1$ for $m,k ge 3$, provide a general lower bound for $text{ESO}(m,k)$, and determine $text{ESO}(m,3)$ up to an additive constant.
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We provide a cyclic permutation analogue of the ErdH os-Szekeres theorem. In particular, we show that every cyclic permutation of length $(k-1)(ell-1)+2$ has either an increasing cyclic sub-permutation of length $k+1$ or a decreasing cyclic sub-permutation of length $ell+1$, and show that the result is tight. We also characterize all maximum-length cyclic permutations that do not have an increasing cyclic sub-permutation of length $k+1$ or a decreasing cyclic sub-permutation of length $ell+1$.
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