We investigate the following Ramsey-type problem. Given a natural number k, determine the smallest integer rr(k) such that, if n is sufficiently large with respect to k, and S is any set of n points in general position in the plane, then all but at most rr(k) points of S can 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.
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics