Satisfiability and computing van der Waerden numbers.
Set-polynomials and polynomial extension of the Hales-Jewett theorem.
Set-systems with restricted multiple intersections.
Some combinatorics involving ξ-large sets
We prove a version of the Ramsey theorem for partitions of (increasing) n-tuples. We derive this result from a version of König's infinity lemma for ξ-large trees. Here ξ < ε₀ and the notion of largeness is in the sense of Hardy hierarchy.
Some Geometric Applications of Dilworth's Theorem.
Some new exact van der Waerden numbers.
Some new Ramsey colorings.
Some Ramsey type theorems for normed and quasinormed spaces
We prove that every bounded, uniformly separated sequence in a normed space contains a “uniformly independent” subsequence (see definition); the constants involved do not depend on the sequence or the space. The finite version of this result is true for all quasinormed spaces. We give a counterexample to the infinite version in for each 0 < p < 1. Some consequences for nonstandard topological vector spaces are derived.
Symmetric sum-free partitions and lower bounds for Schur numbers.