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Školska 2025/26. / Academic year 2025/26 12/05/2026 (Prolećni festival matematike 2026) Jing Zhang (University of North Texas, Denton, USA): A topological rainbow Ramsey theorem Rainbow Ramsey theorem (sometimes called sub-Ramsey theorem) is a weakening of the usual Ramsey theorem which states for any coloring of pairs of natural numbers such that each color is used not too many times, there exists a restriction of the coloring on pairs of an infinite subset which is injective. Unlike Ramsey theorem failing outright for uncountable sets as shown by Sierpinski, Rainbow Ramsey theorem for uncountable sets can consistently hold, shown by Todorcevic in 1983. Topological Ramsey theory is a study on the strengthening of the usual Ramsey theory that imposes additional topological requirements on the target set. In the mid 2000's, Abraham, Cummings and Smyth showed that the following topological rainbow Ramsey theorem is consistent: for any coloring on pairs of the second uncountable cardinal where each color is used at most finitely many times, there exists an uncountable set closed in the order topology on which the restriction of the coloring is injective. Later, Garti and Zhang showed that the boundedness requirement can be relaxed so that each color is used at most countably many times, at the cost of a weaker conclusion: the target set is only stationary in its supremum. We demonstrate a joint extension of the Abraham-Cummings-Smyth theorem and Garti-Zhang theorem by showing that it is indeed consistent to start with an countably-bounded coloring while still insisting the target set is closed in the order topology. The proof involves analysis of a family of games analogous to those usually associated with Chang's conjecture. The talk will be self-contained. This is joint work with Hannes Jakob. 21/01/2026 (Obeležavanje Svetskog dana logike 2026) Ilijas Farah (York Univeristy, Toronto, Canada): Conjugacy of trivial autohomeomorphisms of $\beta N\setminus N$ An autohomeomorphism of the Čech-Stone remainder $\beta N\setminus N$ is called trivial if it has a continuous extension to a map from $\beta N$ into itself. Such map is determined by a bijection between cofinite subsets of $N$, so-called almost permutation. By results of W.Rudin and S.Shelah, the question whether nontrivial autohomeomorphisms exist is independent from ZFC. We will be considering rotary autohomeomorphisms, the ones that correspond to permutations of $N$ all of whose cycles are finite. If all autohomeomorphisms are trivial, then the problem of their conjugacy is also trivial (in the usual sense of the word). If the Continuum Hypothesis holds, then in the rotary case this question has a model-theoretic reformulation. While our results are regrettably incomplete, they suffice to decide whether for example the rotary autohomeomorphisms whose cycles have lengths $2^{2n}$, for $n\in N$, and $2^{2n+1}$, for $n\in N$, are conjugate. This is a joint work with Will Brian. 04/12/2025 (Dani matematike u Novom Sadu 2025) Igor Dolinka: Canone Inverso Petar Marković: Velika prevara - O najtežem problemu koji se rešava u polinomnom vremenu |
