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1. Ramsey numbers of hypergraphs with a given size

   https://doi.org/10.1017/S0305004124000409  

   BRADAČ, DOMAGOJ; FOX, JACOB; SUDAKOV, BENNY (January 2025, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract Theq-colour Ramsey number of ak-uniform hypergraphHis the minimum integerNsuch that anyq-colouring of the completek-uniform hypergraph onNvertices contains a monochromatic copy ofH. The study of these numbers is one of the central topics in Combinatorics. In 1973, Erdős and Graham asked to maximise the Ramsey number of a graph as a function of the number of its edges. Motivated by this problem, we study the analogous question for hypergaphs. For fixed$$k \ge 3$$and$$q \ge 2$$we prove that the largest possibleq-colour Ramsey number of ak-uniform hypergraph withmedges is at most$$\mathrm{tw}_k(O(\sqrt{m})),$$where tw denotes the tower function. We also present a construction showing that this bound is tight for$$q \ge 4$$. This resolves a problem by Conlon, Fox and Sudakov. They previously proved the upper bound for$$k \geq 4$$and the lower bound for$$k=3$$. Although in the graph case the tightness follows simply by considering a clique of appropriate size, for higher uniformities the construction is rather involved and is obtained by using paths in expander graphs. 

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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. 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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4. Supersaturation of even linear cycles in linear hypergraphs

   https://doi.org/10.1017/S0963548320000206  

   Jiang, Tao; Yepremyan, Liana (September 2020, Combinatorics, Probability and Computing)

   Abstract A classical result of Erdős and, independently, of Bondy and Simonovits [3] says that the maximum number of edges in ann-vertex graph not containingC_2k, the cycle of length 2k, isO(n^1+1/k). Simonovits established a corresponding supersaturation result forC_2k’s, showing that there exist positive constantsC,cdepending only onksuch that everyn-vertex graphGwithe(G)⩾Cn^1+1/kcontains at leastc(e(G)/v(G))^2kcopies ofC_2k, this number of copies tightly achieved by the random graph (up to a multiplicative constant). In this paper we extend Simonovits' result to a supersaturation result ofr-uniform linear cycles of even length inr-uniform linear hypergraphs. Our proof is self-contained and includes ther= 2 case. As an auxiliary tool, we develop a reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest. 

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5. Improved Bounds on a Generalization of Tuza's Conjecture

   https://doi.org/10.37236/10829  

   Basit, Abdul; McGinnis, Daniel; Simmons, Henry; Sinnwell, Matt; Zerbib, Shira (October 2022, The Electronic Journal of Combinatorics)

   For an $$r$$-uniform hypergraph $$H$$, let $$\nu^{(m)}(H)$$ denote the maximum size of a set $$M$$ of edges in $$H$$ such that every two edges in $$M$$ intersect in less than $$m$$ vertices, and let $$\tau^{(m)}(H)$$ denote the minimum size of a collection $$C$$ of $$m$$-sets of vertices such that every edge in $$H$$ contains an element of $$C$$. The fractional analogues of these parameters are denoted by $$\nu^{*(m)}(H)$$ and $$\tau^{*(m)}(H)$$, respectively. Generalizing a famous conjecture of Tuza on covering triangles in a graph, Aharoni and Zerbib conjectured that for every $$r$$-uniform hypergraph $$H$$, $$\tau^{(r-1)}(H)/\nu^{(r-1)}(H) \leq \lceil{\frac{r+1}{2}}\rceil$$. In this paper we prove bounds on the ratio between the parameters $$\tau^{(m)}$$ and $$\nu^{(m)}$$, and their fractional analogues. Our main result is that, for every $$r$$-uniform hypergraph~$$H$$,\[ \tau^{*(r-1)}(H)/\nu^{(r-1)}(H) \le \begin{cases} \frac{3}{4}r - \frac{r}{4(r+1)} &\text{for }r\text{ even,}\\\frac{3}{4}r - \frac{r}{4(r+2)} &\text{for }r\text{ odd.} \\\end{cases} \]This improves the known bound of $$r-1$.We also prove that, for every $$r$$-uniform hypergraph $$H$$, $$\tau^{(m)}(H)/\nu^{*(m)}(H) \le \operatorname{ex}_m(r, m+1)$$, where the Turán number $$\operatorname{ex}_r(n, k)$$ is the maximum number of edges in an $$r$$-uniform hypergraph on $$n$$ vertices that does not contain a copy of the complete $$r$$-uniform hypergraph on $$k$$ vertices. Finally, we prove further bounds in the special cases $(r,m)=(4,2)$ and $(r,m)=(4,3)$. 

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