Schur's colouring theorem for non-commuting pairs

Abstract

For G a finite non-Abelian group we write c(G) for the probability that two randomly chosen elements commute and k(G) for the largest integer such that any k(G)-colouring of G is guaranteed to contain a monochromatic quadruple (x,y,xy,yx) with xy not equal to yx. We show that c(G) tends to 0 if and only if k(G) tends to infinity.

7pp; corrected typos

References in corpus (3)

• On one problem of Gowers
• Solving xz=yy in certain subsets of finite groups
• On a two-dimensional analog of Szemeredi's Theorem in Abelian groups

Cited by in corpus (2)

• Complete type amalgamation for non-standard finite groups
• Revisiting sums and products in countable and finite fields