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Tuesday, 25 March 2025 - 10:00 to 11:30 <br />
Place: IM, konírna
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Speaker: Dragan Mašulović, University of Novi Sad<br />
Title: Big Ramsey combinatorics of the Cantor set and a simple proof of the Blass perfect set theorem
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Abstract <br />
<p>The Blass perfect set theorem states that for every positive integer n, every perfect set P of the reals and every finite Borel coloring of n-tuples of elements of P there is a perfect set Q contained in P such that n-tuples of Q attain at most (n - 1)! colors. In essence, this is a statement about the Cantor set having finite big Ramsey degrees.</p><p><br></p><p> The original 1981 proof which relies on the Halpern-Läuchli theorem was later simplified by Todorčević who used Miliken spaces to obtain the same result. In this talk we will outline a much simpler proof of the Blass perfect set theorem based on the Carlson-Simpson theorem. We show that for every positive integer n there is a positive integer t such that for every finite Borel coloring of n-tuples of the Cantor set C there is a Cantor set C' contained in C such that n-tuples of C' attain at most t colors. The price we have to pay for the simpler proof is the fact that using less elaborate strategies we are no longer able to show that t = (n - 1)!.</p><p><br></p><p> We conclude the talk with a few remarks showing that several structures related to the Cantor set also have finite big Ramsey degrees.</p>
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For more information see the seminar web page at <br />
https://www.math.cas.cz/index.php/events/seminar/6
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Set Theory and Analysis mailing list <br />
settfa@math.cas.cz <br />
https://list.math.cas.cz/listinfo/settfa@math.cas.cz
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