Tutte seminar - Konstantinos Tyros

Density Ramsey Type Results

Abstract: 

In this talk we will present the density versions of the Hales--Jewett Theorem and the Carlson--Simpson Theorem. The Hales--Jewett Theorem is one of the most representing theorems in Ramsey theory (see \cite{HJ}). Its density version was first  proved by H. Furstenberg and Y. Katznelson in 1991 using Ergodic Theory (see \cite{FK2}). However, since then, combinatorial proofs have been discovered (see \cite{Pol} and \cite{DKT3}). We will comment on these proofs. The Density Carlson--Simpson Theorem is an extension of the Density Hales-Jewett Theorem and concerns the space of the left variable words. The proofs of the above results required a new regularity method that led to a concentration inequality which we will present if time permits.

 \begin{thebibliography}{99}
    \bibitem{CS} T. J. Carlson and S. G. Simpson, \textit{A dual form of Ramsey's theorem}, Adv. Math., 53 (1984), 265-290.
    \bibitem{DKT3} P. Dodos, V. Kanellopoulos and K. Tyros, \textit{A simple proof of the density Hales--Jewett theorem}, International Mathematical Research Notices, to appear.
    \bibitem{DKT} P. Dodos, V. Kanellopoulos and K. Tyros, \textit{A density version of the Carlson--Simpson theorem}, Journal of the European Mathematical Society, to appear.
    \bibitem{FK2} H. Furstenberg and Y. Katznelson, \textit{A density version of the Hales--Jewett theorem}, Journal d'Anal. Math., 57 (1991), 64-119.
    \bibitem{HJ} A. H. Hales and R. I. Jewett, \textit{Regularity and positional games}, Trans. Amer. Math. Soc., 106 (1963), 222-229.
    \bibitem{Pol} D. H. J. Polymath, \textit{A new proof of the density Hales--Jewett theorem}, Ann. Math., 175 (2012), 1283-1327.
  \end{thebibliography}

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