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1. A remark on the geography problem in Heegaard Floer homology

   https://doi.org/10.1090/pspum/102/08  

   Hanselman, Jonathan; Kutluhan, Çagatay; Lidman, Tye (July 2019, Proceedings of symposia in pure mathematics)

   Gay, David; Wu, Weiwei (Ed.)

   We give new obstructions to the module structures arising in Heegaard Floer homology. As a corollary, we characterize the possible modules arising as the Heegaard Floer homology of an integer homology sphere with one-dimensional reduced Floer homology. Up to absolute grading shifts, there are only two. We use this corollary to show that the chain complex depicted by Ozsváth, Stipsicz, and Szabó to argue that there is no algebraic obstruction to the existence of knots with trivial epsilon invariant and non-trivial upsilon invariant cannot be realized as the knot Floer complex of a knot. 

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2. A combinatorial description of the LOSS Legendrian knot invariant

   https://doi.org/10.1142/S0218216522500936  

   He, Dongtai; Truong, Linh (November 2022, Journal of Knot Theory and Its Ramifications)

   In this paper, we observe that the hat version of the Heegaard Floer invariant of Legendrian knots in contact three-manifolds defined by Lisca-Ozsváth-Stipsicz-Szabó can be combinatorially computed. We rely on Plamenevskaya’s combinatorial description of the Heegaard Floer contact invariant. 

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3. Four-dimensional aspects of tight contact 3-manifolds

   https://doi.org/10.1073/pnas.2025436118  

   Hedden, Matthew; Raoux, Katherine (May 2021, Proceedings of the National Academy of Sciences)

   We conjecture a four-dimensional characterization of tightness: A contact structure on a 3-manifold Y is tight if and only if a slice-Bennequin inequality holds for smoothly embedded surfaces in $Y\times \left[\mathrm{0,1}\right]$. An affirmative answer to our conjecture would imply an analogue of the Milnor conjecture for torus knots: If a fibered link L induces a tight contact structure on Y, then its fiber surface maximizes the Euler characteristic among all surfaces in $Y\times \left[\mathrm{0,1}\right]$with boundary L. We provide evidence for both conjectures by proving them for contact structures with nonvanishing Ozsváth–Szabó contact invariant. 

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4. Floer Homology Beyond Borders

   Lipshitz, Robert; Ozsváth, Peter; Thurston, Dylan (September 2024, International Press)

   Gross, David; Yao, Andrew Chi-Chih; Yau, Shing-Tung (Ed.)

   Bordered Floer homology is an invariant for 3-manifolds with boundary, defined by the authors in 2008. It extends the Heegaard Floer homology of closed 3-manifolds, defined in earlier work of Zoltán Szabó and the second author. In addition to its conceptual interest, bordered Floer homology also provides powerful computational tools. This survey outlines the theory, focusing on recent developments and applications. 

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5. Multicolor list Ramsey numbers grow exponentially

   https://doi.org/10.1002/jgt.22832  

   Fox, Jacob; He, Xiaoyu; Luo, Sammy; Xu, Max Wenqiang (April 2022, Journal of Graph Theory)

   Abstract The list Ramsey number , recently introduced by Alon, Bucić, Kalvari, Kuperwasser, and Szabó, is a list‐coloring variant of the classical Ramsey number. They showed that if is a fixed ‐uniform hypergraph that is not ‐partite and the number of colors goes to infinity, . We prove that if and only if is not ‐partite. 

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