Erdős-Szekeres On-Line

In 1935, Erdős 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 Erdős-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.