Abstract
We investigate the following Ramsey-type problem. Given a natural number k,determine the smallest integer rr(k)such that, if nis sufficiently large with respect to k,and Sis any set of npoints in general position in the plane, then all but at most rr(k)points of Scan be partitioned into convex sets of sizes ⩾ k.We provide estimates on rr(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 number r(k, k-1) minus 1.
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