Turán-type results for partial orders and intersection graphs of convex sets
نویسندگان
چکیده
We prove Ramsey-type results for intersection graphs of geometric objects the plane. In particular, we prove the following bounds, all of which are tight apart from the constant c. There is a constant c > 0 such that for every family F of n ≥ 2 convex sets in the plane, the intersection graph of F or its complement contains a balanced complete bipartite graph of size at least cn. There is a constant c′ > 0 such that for every family F of x-monotone curves in the plane, the intersection graph G of F contains a balanced complete bipartite graph of size at least cn/ log n or the complement of G contains a balanced complete bipartite graph of size at least cn. Our bounds rely on new Turán-type results on incomparability graphs of partially ordered sets.
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