Algebraic Ramsey-theoretic results with small monochromatic sets

יום ב', 05/11/2018 - 13:00
We will explore some (recent and not so recent; some positive, some negative) Ramsey-type results (each of which is due to some subset of the set {Komj\'ath, Hindman, Leader, H.S. Lee, P. Russell, Shelah, D. Soukup, Strauss, Rinot, Vidnyánszky, myself}) where abelian groups are coloured, and one attempts to obtain monochromatic sets defined in terms of the group structure. We will focus specifically on two families of very recent results: the first one concerns colouring groups with uncountably many colours, attempting to obtain finite monochromatic FS-sets; the second one concerns colouring groups (most of the time, our group of interest is the real line $\mathbb R$ with its usual addition) with finitely many colours, attempting to obtain countably infinite monochromatic sumsets.