Some unifying principles in Ramsey theory (Q1813078)

A space of infinite sequences is a Ramsey space if every set which has the property of Baire is Ramsey and no open set is meager. The author considers special structures \({\mathcal S}(L,{\mathbf e})\), \({\mathcal S}^ +(L, {\mathbf e})\) (\(L\) being an alphabet, \({\mathbf e}\) an infinite sequence of natural numbers). The main result of this paper lies in characterizing those situations when \({\mathcal S}(L, {\mathbf e})\) and \({\mathcal S}^ +(L, {\mathbf e})\) are Ramsey spaces. This approach provides among other results a generalization of a large part of the qualitative Ramsey theory. Several results are derived from the fact that \({\mathcal S}(L, {\mathbf e})\) is a Ramsey space for finite \(L\) and \({\mathbf e}\) constantly equal to 1; examples are Ellentuck's theorem [\textit{E. Ellentuck}, J. Symb. Logic 39, 163--165 (1974; Zbl 0292.02054)], Milliken's extension of Hindman's theorem [\textit{K. R. Milliken}, J. Comb. Theory, Ser. A 18, 276--290 (1975; Zbl 0323.05001)], the dual of Ellentuck's theorem [the author and \textit{S. G. Simpson}, Adv. Math. 53, 265--290 (1984; Zbl 0564.05005)], the Hales-Jewett theorem [\textit{A. W. Hales} and \textit{R. I. Jewett}, Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] and the Graham-Rothschild theorem on \(n\)-parameter sets [\textit{R. L. Graham} and \textit{B. L. Rothschild}, ibid. 159, 257--292 (1971; Zbl 0233.05003)].