Some Recent Results in Ramsey Theory

We review and comment on a number of results in Ramsey theory obtained recently by the author in collaboration with V. Kanellopoulos, N. Karagiannis and K. Tyros. Among them are density versions of the classical pigeonhole principles of Halpern–Läuchli and Carlson–Simpson. We shall comment on recent progress concerning one fundamental problem in Ramsey theory. It originates from an insightful conjecture of Erdős and Turán [20] and, in full generality, asks to determine which pigeonhole principles admit a density version. There have been numerous dramatic developments in this direction—see, e.g., [4, 26, 27, 39, 46] and the references therein—which go well beyond the scope of the present review. We will thus be forced to neglect a vast amount of remarkable current research. We shall focus, instead, on three basic pigeonhole principles which are particularly appealing due to their widespread utility and unifying power. 1. The coloring versions 1.1. The first pigeonhole principle relevant to our discussion is the Hales–Jewett theorem [28]. To state it we need to introduce some pieces of notation and some terminology. For every integer k > 2 let [k] 2 and r > 1 there exists a positive integer N with the following property. If n > N , then for every r-coloring of [k] there exists a combinatorial line of [k] which is monochromatic. The least positive integer with this property will be denoted by HJ(k, r). 2000 Mathematics Subject Classification: 05D10.