A matrix A is said to be partition regular (PR) over a subset S of the positive integers if whenever S is finitely coloured, there exists a vector x, with all elements in the sam...
Let G be a d-regular graph with girth g, and let α be the independence number of G. We show that α(G) ≥ 1 2 1 − (d − 1)−2/(d−2) − (g) n where (g) → 0 as g → ∞,...
We study thresholds for Ramsey properties of random discrete structures. In particular, we determine the threshold for Rado’s theorem for solutions of partition regular systems o...
A linear equation L is called k-regular if every k-coloring of the positive integers contains a monochromatic solution to L. Richard Rado conjectured that for every positive intege...
It is easy to deduce from Ramsey's Theorem that, given positive integers a1, a2, . . . , am and a finite colouring of the set N of positive integers, there exists an injectiv...