sufficiently large structure we can find a homogeneous substructure of
nontrivial size. Actually, one of the first results in this field was
in geometric context. In 1935 Erdos and Szekeres proved that among n^2
points in a plane n of them form a monotone subset, and that every set
of 2^n points contains an n-point convex or an n-point concave subset.
Recently, there was a progress in this area concerning Semialgebraic
predicates. Conlon, Fox, Pach, Sudakov, and Suk showed that every set of
twr_k(n^c) points in R^d equipped with a (k+1)-ary semialgebraic
predicate contains an n-point set homogeneous according to this
predicate. They also constructed a set S of twr_k(cn) points in
dimension 2^{k-2} and a (k+1)-ary semialgebraic predicate P such that
there is no P-homogeneous subset of S of size n+1. In this talk we
present a result with Matousek, Roldan-Pensado, and Safernova which
lowers the dimension needed for this construction, as well as bounds and
constructions for some specific predicates e.g. Order type.
joint work with J. Matousek, E. Roldan-Pensado, and Z. Safernova