The automorphism group of the random lattice is not amenable
We prove that the automorphism group of the random lattice is not amenable, and we identify the universal minimal flow for the automorphism group of the random distributive lattice.
The minimum number of monotone subsequences.
The wildcard set's size in the density Hales-Jewett theorem.
Thue-like sequences and rainbow arithmetic progressions.
Triangle free sets and arithmetic progressions---two Pisier type problems.
Two color off-diagonal Rado-type numbers.
Two new extensions of the Hales-Jewett theorem.
Two variants of the size Ramsey number
Given a graph H and an integer r ≥ 2, let G → (H,r) denote the Ramsey property of a graph G, that is, every r-coloring of the edges of G results in a monochromatic copy of H. Further, let and define the Ramsey density as the infimum of m(G) over all graphs G such that G → (H,r). In the first part of this paper we show that when H is a complete graph Kₖ on k vertices, then , where R = R(k;r) is the classical Ramsey number. As a corollary we derive a new proof of the result credited to Chvatál...