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1. Generalizations and Strengthenings of Ryser's Conjecture

   https://doi.org/10.37236/9914  

   DeBiasio, Louis; Kamel, Yigal; McCourt, Grace; Sheats, Hannah (October 2021, The Electronic Journal of Combinatorics)

   Ryser's conjecture says that for every $$r$$-partite hypergraph $$H$$ with matching number $$\nu(H)$$, the vertex cover number is at most $$(r-1)\nu(H)$$.  This far-reaching generalization of König's theorem is only known to be true for $$r\leq 3$$, or when $$\nu(H)=1$$ and $$r\leq 5$$.  An equivalent formulation of Ryser's conjecture is that in every $$r$$-edge coloring of a graph $$G$$ with independence number $$\alpha(G)$$, there exists at most $$(r-1)\alpha(G)$$ monochromatic connected subgraphs which cover the vertex set of $$G$$.   We make the case that this latter formulation of Ryser's conjecture naturally leads to a variety of stronger conjectures and generalizations to hypergraphs and multipartite graphs.  Regarding these generalizations and strengthenings, we survey the known results, improving upon some, and we introduce a collection of new problems and results. 

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2. On Off-Diagonal Hypergraph Ramsey Numbers

   https://doi.org/10.1093/imrn/rnaf122  

   Conlon, David; Fox, Jacob; Gunby, Benjamin; He, Xiaoyu; Mubayi, Dhruv; Suk, Andrew; Verstraëte, Jacques (June 2025, International Mathematics Research Notices)

   Abstract A fundamental problem in Ramsey theory is to determine the growth rate in terms of $$n$$ of the Ramsey number $$r(H, K_{n}^{(3)})$$ of a fixed $$3$$-uniform hypergraph $$H$$ versus the complete $$3$$-uniform hypergraph with $$n$$ vertices. We study this problem, proving two main results. First, we show that for a broad class of $$H$$, including links of odd cycles and tight cycles of length not divisible by three, $$r(H, K_{n}^{(3)}) \ge 2^{\Omega _{H}(n \log n)}$$. This significantly generalizes and simplifies an earlier construction of Fox and He which handled the case of links of odd cycles and is sharp both in this case and for all but finitely many tight cycles of length not divisible by three. Second, disproving a folklore conjecture in the area, we show that there exists a linear hypergraph $$H$$ for which $$r(H, K_{n}^{(3)})$$ is superpolynomial in $$n$$. This provides the first example of a separation between $$r(H,K_{n}^{(3)})$$ and $$r(H,K_{n,n,n}^{(3)})$$, since the latter is known to be polynomial in $$n$$ when $$H$$ is linear. 

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3. On the upper tail problem for random hypergraphs

   https://doi.org/10.1002/rsa.20975  

   Liu, Yang P.; Zhao, Yufei (November 2020, Random Structures & Algorithms)

   The upper tail problem in a random graph asks to estimate the probability that the number of copies of some fixed subgraph in an Erdős‐Rényi random graph exceeds its expectation by some constant factor. There has been much exciting recent progress on this problem. We study the corresponding problem for hypergraphs, for which less is known about the large deviation rate. We present new phenomena in upper tail large deviations for sparse random hypergraphs that are not seen in random graphs. We conjecture a formula for the large deviation rate, that is, the first order asymptotics of the log‐probability that the number of copies of fixed subgraphHin a sparse Erdős‐Rényi randomk‐uniform hypergraph exceeds its expectation by a constant factor. This conjecture turns out to be significantly more intricate compared to the case for graphs. We verify our conjecture when the fixed subgraphHbeing counted is a clique, as well as whenHis the 3‐uniform 6‐vertex 4‐edge hypergraph consisting of alternating faces of an octahedron, where new techniques are required. 

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4. Tight paths in convex geometric hypergraphs

   https://doi.org/10.19086/aic.12044  

   Mubayi, Dhruv; Füredi, Zoltán; Verstraëte, Jacques; Kostochka, Alexandr; Jiang, Tao (February 2020, Advances in Combinatorics)

   One of the most intruguing conjectures in extremal graph theory is the conjecture of Erdős and Sós from 1962, which asserts that every $$n$$-vertex graph with more than $$\frac{k-1}{2}n$$ edges contains any $$k$$-edge tree as a subgraph. Kalai proposed a generalization of this conjecture to hypergraphs. To explain the generalization, we need to define the concept of a tight tree in an $$r$$-uniform hypergraph, i.e., a hypergraph where each edge contains $$r$$ vertices. A tight tree is an $$r$$-uniform hypergraph such that there is an ordering $$v_1,\ldots,v_n$$ of its its vertices with the following property: the vertices $$v_1,\ldots,v_r$$ form an edge and for every $i>r$, there is a single edge $$e$$ containing the vertex $$v_i$$ and $r-1$ of the vertices $$v_1,\ldots,v_{i-1}$$, and $$e\setminus\{v_i\}$$ is a subset of one of the edges consisting only of vertices from $$v_1,\ldots,v_{i-1}$$. The conjecture of Kalai asserts that every $$n$$-vertex $$r$$-uniform hypergraph with more than $$\frac{k-1}{r}\binom{n}{r-1}$$ edges contains every $$k$$-edge tight tree as a subhypergraph. The recent breakthrough results on the existence of combinatorial designs by Keevash and by Glock, Kühn, Lo and Osthus show that this conjecture, if true, would be tight for infinitely many values of $$n$$ for every $$r$$ and $$k$$.The article deals with the special case of the conjecture when the sought tight tree is a path, i.e., the edges are the $$r$$-tuples of consecutive vertices in the above ordering. The case $r=2$ is the famous Erdős-Gallai theorem on the existence of paths in graphs. The case $r=3$ and $k=4$ follows from an earlier work of the authors on the conjecture of Kalai. The main result of the article is the first non-trivial upper bound valid for all $$r$$ and $$k$$. The proof is based on techniques developed for a closely related problem where a hypergraph comes with a geometric structure: the vertices are points in the plane in a strictly convex position and the sought path has to zigzag beetwen the vertices. 

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5. A Hypergraph Analog of Dirac's Theorem for Long Cycles in 2-Connected Graphs, II: Large Uniformities

   https://doi.org/10.37236/12486  

   Kostochka, Alexandr; Luo, Ruth; McCourt, Grace (January 2025, The Electronic Journal of Combinatorics)

   Dirac proved that each $$n$$-vertex $$2$$-connected graph with minimum degree $$k$$ contains a cycle of length at least $$\min\{2k, n\}$$. We obtain analogous results for Berge cycles in hypergraphs. Recently, the authors proved an exact lower bound on the minimum degree ensuring a Berge cycle of length at least $$\min\{2k, n\}$$ in $$n$$-vertex $$r$$-uniform $$2$$-connected hypergraphs when $$k \geq r+2$$. In this paper we address the case $$k \leq r+1$$ in which the bounds have a different behavior. We prove that each $$n$$-vertex $$r$$-uniform $$2$$-connected hypergraph $$H$$ with minimum degree $$k$$ contains a Berge cycle of length at least $$\min\{2k,n,|E(H)|\}$$. If $$|E(H)|\geq n$$, this bound coincides with the bound of the Dirac's Theorem for 2-connected graphs. 

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