The Size of a Hypergraph and its Matching Number

More than forty years ago, Erdős conjectured that for any $t \leq \frac{n}{k}$, every k-uniform hypergraph on n vertices without t disjoint edges has at most max${\binom{kt-1}{k}, \binom{n}{k}-\binom{n-t+1}{k}\}$ edges. Although this appears to be a basic instance of the hypergraph Turán problem (with a t-edge matching as the excluded hypergraph), progress on this question has remained elusive. In this paper, we verify this conjecture for all $t < \frac{n}{3k^2}$. This improves upon the best previously known range $t = O\bigl(\frac{n}{k^3}\bigr)$, which dates back to the 1970s.

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The Turán Problem for Hypergraphs of Fixed Size

The Electronic Journal of Combinatorics ◽

10.37236/1978 ◽

2005 ◽

Vol 12 (1) ◽

Cited By ~ 1

Author(s):

Peter Keevash

Keyword(s):

Uniform Hypergraph ◽

Fixed Size ◽

Turan Problem ◽

General Bound ◽

Turán Problem

We obtain a general bound on the Turán density of a hypergraph in terms of the number of edges that it contains. If ${\cal F}$ is an $r$-uniform hypergraph with $f$ edges we show that $$\pi({\cal F}) < {f-2\over f-1} - \big(1+o(1)\big)(2r!^{2/r}f^{3-2/r})^{-1},$$ for fixed $r \geq 3$ and $f \rightarrow \infty$.

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On a Two-Sided Turán Problem

The Electronic Journal of Combinatorics ◽

10.37236/1735 ◽

2003 ◽

Vol 10 (1) ◽

Cited By ~ 1

Author(s):

Dhruv Mubayi ◽

Yi Zhao

Keyword(s):

Minimum Size ◽

Turan Problem ◽

Turan Problems ◽

Turán Problem ◽

Positive Integers

Given positive integers $n,k,t$, with $2 \le k\le n$, and $t < 2^k$, let $m(n,k,t)$ be the minimum size of a family ${\cal F}$ of nonempty subsets of $[n]$ such that every $k$-set in $[n]$ contains at least $t$ sets from ${\cal F}$, and every $(k-1)$-set in $[n]$ contains at most $t-1$ sets from ${\cal F}$. Sloan et al. determined $m(n, 3, 2)$ and Füredi et al. studied $m(n, 4, t)$ for $t=2, 3$. We consider $m(n, 3, t)$ and $m(n, 4, t)$ for all the remaining values of $t$ and obtain their exact values except for $k=4$ and $t= 6, 7, 11, 12$. For example, we prove that $ m(n, 4, 5) = {n \choose 2}-17$ for $n\ge 160$. The values of $m(n, 4, t)$ for $t=7,11,12$ are determined in terms of well-known (and open) Turán problems for graphs and hypergraphs. We also obtain bounds of $m(n, 4, 6)$ that differ by absolute constants.

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Inverting the Turán problem with chromatic number

Discrete Mathematics ◽

10.1016/j.disc.2021.112517 ◽

2021 ◽

Vol 344 (9) ◽

pp. 112517

Author(s):

Xiutao Zhu ◽

Yaojun Chen

Keyword(s):

Chromatic Number ◽

Turan Problem ◽

Turán Problem

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On a Turán problem in weakly quasirandom 3-uniform hypergraphs

Journal of the European Mathematical Society ◽

10.4171/jems/784 ◽

2018 ◽

Vol 20 (5) ◽

pp. 1139-1159 ◽

Cited By ~ 4

Author(s):

Christian Reiher ◽

Vojtěch Rödl ◽

Mathias Schacht

Keyword(s):

Uniform Hypergraphs ◽

Turan Problem ◽

Turán Problem

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An Ordered Turán Problem for Bipartite Graphs

The Electronic Journal of Combinatorics ◽

10.37236/2471 ◽

2012 ◽

Vol 19 (4) ◽

Author(s):

Craig Timmons

Keyword(s):

Complete Bipartite Graph ◽

Bipartite Graphs ◽

Free Graph ◽

The Family ◽

Turan Problem ◽

Vertex Graph ◽

Turán Number ◽

Turán Problem ◽

Natural Way

Let $F$ be a graph. A graph $G$ is $F$-free if it does not contain $F$ as a subgraph. The Turán number of $F$, written $\textrm{ex}(n,F)$, is the maximum number of edges in an $F$-free graph with $n$ vertices. The determination of Turán numbers of bipartite graphs is a challenging and widely investigated problem. In this paper we introduce an ordered version of the Turán problem for bipartite graphs. Let $G$ be a graph with $V(G) = \{1, 2, \dots , n \}$ and view the vertices of $G$ as being ordered in the natural way. A zig-zag $K_{s,t}$, denoted $Z_{s,t}$, is a complete bipartite graph $K_{s,t}$ whose parts $A = \{n_1 < n_2 < \dots < n_s \}$ and $B = \{m_1 < m_2 < \dots < m_t \}$ satisfy the condition $n_s < m_1$. A zig-zag $C_{2k}$ is an even cycle $C_{2k}$ whose vertices in one part precede all of those in the other part. Write $\mathcal{Z}_{2k}$ for the family of zig-zag $2k$-cycles. We investigate the Turán numbers $\textrm{ex}(n,Z_{s,t})$ and $\textrm{ex}(n,\mathcal{Z}_{2k})$. In particular we show $\textrm{ex}(n, Z_{2,2}) \leq \frac{2}{3}n^{3/2} + O(n^{5/4})$. For infinitely many $n$ we construct a $Z_{2,2}$-free $n$-vertex graph with more than $(n - \sqrt{n} - 1) + \textrm{ex} (n,K_{2,2})$ edges.

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