Regularity and positional games

Abstract approach to finite Ramsey theory and a self-dual Ramsey theorem

“…In the particular case, when U is 0-dimensional and is identified with f : [n] → A, we say that f is a function in V . This translates to f being an extension of g to [n] that is constant on each set in G. In the particular case, when V is a 0-parameter set and, therefore, so is U , and they are both identified with the partitions G and F , respectively, we have that F is coarser than G. The first theorem is from [4]. Note that the Dual Ramsey Theorem is an instance of the Graham-Rothschild Theorem for A = 0 and k, l > 0.…”

“…We prove below a theorem that combines into one the usual Hales-Jewett theorem [4] and Voigt's version of this theorem for partial functions [22,Theorem 2.7] as phrased in Section 2.3. One gets the classical Hales-Jewett theorem from the statement below by setting L = L 0 + 1, L 0 = K 0 , and v 0 = id [K 0 ] in the assumption and L = L in the conclusion.…”

Extremal Combinatorics

“…?? we will show how this theorem can be derived using a very powerful Ramsey-type result due to Hales and Jewett (1963).…”

“…Theorem 26.1 (Hales-Jewett 1963). For every natural numbers t and r there exists a dimension n = HJ(r, t) such that whenever [t] n is r-colored, there exists a monochromatic line.…”

Subsets of products of finite sets of positive upper density

“…Density versions of Ramsey-theoretic results tend to be considerably harder to prove. A typical such example is the density version of the Hales-Jewett theorem [7] due to Furstenberg and Katznelson [5] (see also [9]). All these examples belong to finite Ramsey theory in the sense of [6].…”