Related papers: Constructing Dense Grid-Free Linear $3$-Graphs

We give a new construction showing that for every $r\ge 3$, there exists an $r$-uniform linear hypergraph on $n$ vertices with $\Theta_r(n^2)$ edges and no copy of the $r\times r$ grid. This complements the works of F\"uredi--Ruszink\'o,…

We show a construction for dense 3-uniform linear hypergraphs without $3\times 3$ grids, improving the lower bound on its Tur\'an number.

We prove that the maximum number of edges in a 3-uniform linear hypergraph on $n$ vertices containing no 2-regular subhypergraph is $n^{1+o(1)}$. This resolves a conjecture of Dellamonica, Haxell, Luczak, Mubayi, Nagle, Person, R\"odl,…

We study the maximum number of hyperedges in a 3-uniform hypergraph on $n$ vertices that does not contain a Berge cycle of a given length $\ell$. In particular we prove that the upper bound for $C_{2k+1}$-free hypergraphs is of the order…

Gy\'arf\'as, Gy\H{o}ri and Simonovits proved that if a $3$-uniform hypergraph with $n$ vertices has no linear cycles, then its independence number $\alpha \ge \frac{2n} {5}$. The hypergraph consisting of vertex disjoint copies of a complete…

In this paper we show that the maximum number of hyperedges in a $3$-uniform hypergraph on $n$ vertices without a (Berge) cycle of length five is less than $(0.254 + o(1))n^{3/2}$, improving an estimate of Bollob\'as and Gy\H{o}ri. We…

Extensions of Erd\H{o}s-Gallai Theorem for general hypergraphs are well studied. In this work, we prove the extension of Erd\H{o}s-Gallai Theorem for linear hypergraphs. In particular, we show that the number of hyperedges in an $n$-vertex…

The conjecture of Brown, Erd\H{o}s and S\'os from 1973 states that, for any $k \ge 3$, if a $3$-uniform hypergraph $H$ with $n$ vertices does not contain a set of $k+3$ vertices spanning at least $k$ edges then it has $o(n^2)$ edges. The…

Erd\H{o}s conjectured that every $n$-vertex triangle-free graph contains a subset of $\lfloor n/2\rfloor$ vertices that spans at most $n^2/50$ edges. Extending a recent result of Norin and Yepremyan, we confirm this conjecture for graphs…

In this note we show that the maximum number of edges in a $3$-uniform hypergraph without a Berge cycle of length four is at most $(1+o(1))\frac{n^{3/2}}{\sqrt{10}}$. This improves earlier estimates by Gy\H{o}ri and Lemons and by F\"uredi…

The famous Brown-Erd\H{o}s-S\'os conjecture from 1973 states, in an equivalent form, that for any fixed $\delta>0$ and integer $k\geq 3$ every sufficiently large linear $3$-uniform hypergraph of size $\delta n^2$ contains some $k$ edges…

In this note, we prove several Tur\'an-type results on geometric hypergraphs. The two main theorems are 1) Every $n$-vertex geometric 3-hypergraph in 2-space with no three strongly crossing edges has at most $O(n^2)$ edges, 2) Every…

Let $D_2$ denote the $3$-uniform hypergraph with $4$ vertices and $2$ edges. Answering a question of Alon and Shapira, we prove an induced removal lemma for $D_2$ having polynomial bounds. We also prove an Erd\H{o}s-Hajnal-type result:…

Let $Y_{3,2}$ be the 3-graph with two edges intersecting in two vertices. We prove that every 3-graph $ H $ on $ n $ vertices with at least $ \max \left \{ \binom{4\alpha n}{3}, \binom{n}{3}-\binom{n-\alpha n}{3} \right \}+o(n^3) $ edges…

An $r$-uniform tight cycle of length $\ell>r$ is a hypergraph with vertices $v_1,\dots,v_\ell$ and edges $\{v_i,v_{i+1},\dots,v_{i+r-1}\}$ (for all $i$), with the indices taken modulo $\ell$. It was shown by Sudakov and Tomon that for each…

In this paper, we study the problem of determining the maximum number of edges in an $n$-vertex $r$-uniform hypergraph that contains no $(k+1)$-connected subgraph. The graph case is a classical problem initiated by Mader, central to graph…

We explore properties of $3$-uniform hypergraphs $H$ without linear cycles. Our main results are that these hypergraphs must contain a vertex of strong degree at most two and must have independent sets of size at least ${2|V(H)|\over 5}$.

We prove the well-known Brown-Erd\H{o}s-S\'os Conjecture for hypergraphs of large uniformity in the following form: any dense linear $r$-graph $G$ has $k$ edges spanning at most $(r-2)k+3$ vertices, provided the uniformity $r$ of $G$ is…

For a 3-uniform hypergraph (3-graph) $F$, let $r(F,n)$ be the smallest $N$ such that any $N$-vertex $F$-free 3-graph has an independent set of size $n$. We construct a $3$-graph $H_2$ with six vertices and five edges such that…

A hypergraph is called an r by r grid if it is isomorphic to a pattern of r horizontal and r vertical lines. Three sets form a triangle if they pairwise intersect in three distinct singletons. A hypergraph is linear if every pair of edges…