Relative Turán Problems for Uniform Hypergraphs

Sam Spiro; Jacques Verstraëte

doi:10.1137/20m1364631

Turán Problems on Non-Uniform Hypergraphs

The Electronic Journal of Combinatorics ◽

10.37236/3901 ◽

2014 ◽

Vol 21 (4) ◽

Cited By ~ 5

Author(s):

J. Travis Johnston ◽

Linyuan Lu

Keyword(s):

Recent Work ◽

Blow Up ◽

Uniform Hypergraph ◽

Expected Number ◽

Uniform Hypergraphs ◽

Turan Problems ◽

Vertex Set ◽

Lubell Function

A non-uniform hypergraph $H=(V,E)$ consists of a vertex set $V$ and an edge set $E\subseteq 2^V$; the edges in $E$ are not required to all have the same cardinality. The set of all cardinalities of edges in $H$ is denoted by $R(H)$, the set of edge types. For a fixed hypergraph $H$, the Turán density $\pi(H)$ is defined to be $\lim_{n\to\infty}\max_{G_n}h_n(G_n)$, where the maximum is taken over all $H$-free hypergraphs $G_n$ on $n$ vertices satisfying $R(G_n)\subseteq R(H)$, and $h_n(G_n)$, the so called Lubell function, is the expected number of edges in $G_n$ hit by a random full chain. This concept, which generalizes the Turán density of $k$-uniform hypergraphs, is motivated by recent work on extremal poset problems. The details connecting these two areas will be revealed in the end of this paper.Several properties of Turán density, such as supersaturation, blow-up, and suspension, are generalized from uniform hypergraphs to non-uniform hypergraphs. Other questions such as "Which hypergraphs are degenerate?" are more complicated and don't appear to generalize well. In addition, we completely determine the Turán densities of $\{1,2\}$-hypergraphs.

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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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Rainbow matchings for 3-uniform hypergraphs

Journal of Combinatorial Theory Series A ◽

10.1016/j.jcta.2021.105489 ◽

2021 ◽

Vol 183 ◽

pp. 105489

Author(s):

Hongliang Lu ◽

Xingxing Yu ◽

Xiaofan Yuan

Keyword(s):

Uniform Hypergraphs

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Hypercycle Systems of 5-Cycles in Complete 3-Uniform Hypergraphs

Mathematics ◽

10.3390/math9050484 ◽

2021 ◽

Vol 9 (5) ◽

pp. 484

Author(s):

Anita Keszler ◽

Zsolt Tuza

Keyword(s):

Element Subset ◽

Uniform Hypergraphs ◽

Vertex Set ◽

Recursive Constructions

In this paper, we consider the problem of constructing hypercycle systems of 5-cycles in complete 3-uniform hypergraphs. A hypercycle system C(r,k,v) of order v is a collection of r-uniform k-cycles on a v-element vertex set, such that each r-element subset is an edge in precisely one of those k-cycles. We present cyclic hypercycle systems C(3,5,v) of orders v=25,26,31,35,37,41,46,47,55,56, a highly symmetric construction for v=40, and cyclic 2-split constructions of orders 32,40,50,52. As a consequence, all orders v≤60 permitted by the divisibility conditions admit a C(3,5,v) system. New recursive constructions are also introduced.

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