Ramsey-remainder

We investigate the following Ramsey-type problem. Given a natural numberk,determine the smallest integerrr(k)such that, ifnis sufficiently large with respect tok,andSis any set ofnpoints in general position in the plane, then all but at mostrr(k)points ofScan be partitioned into convex sets of sizes ⩾ k.We provide estimates onrr(k)which are best possible if a classic conjecture of Erdos and Szekeres on the Ramsey number for convex sets is valid. We also prove that in several types of combinatorial structures, the corresponding ‘Ramsey-remainder’rr(k)is equal to the off-diagonal Ramsey numberr(k, k-1)minus 1.