scholarly journals Rational exponents for hypergraph Turan problems

2019 ◽  
Vol 10 (1) ◽  
pp. 61-86
Author(s):  
Matthew Fitch
Keyword(s):  
Turan Problems

On a Two-Sided Turán Problem

10.37236/1735 ◽  
2003 ◽  
Vol 10 (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.

Relative Turán Problems for Uniform Hypergraphs

2021 ◽  
Vol 35 (3) ◽  
pp. 2170-2191
Author(s):  
Sam Spiro ◽  
Jacques Verstraëte
Keyword(s):  
Uniform Hypergraphs ◽  
Turan Problems

scholarly journals Some tight lower bounds for Turán problems via constructions of multi-hypergraphs

2020 ◽  
Vol 89 ◽  
pp. 103161
Author(s):  
Zixiang Xu ◽  
Tao Zhang ◽  
Gennian Ge
Keyword(s):  
Lower Bounds ◽  
Turan Problems

Turán problems for vertex-disjoint cliques in multi-partite hypergraphs

2020 ◽  
Vol 343 (10) ◽  
pp. 112005
Author(s):  
Erica L.L. Liu ◽  
Jian Wang
Keyword(s):  
Turan Problems ◽  
Vertex Disjoint

On supersaturation and stability for generalized Turán problems

2020 ◽  
Author(s):  
Anastasia Halfpap ◽  
Cory Palmer
Keyword(s):  
Turan Problems

Hypergraph Turán problems

2011 ◽  
pp. 83-140 ◽  
Author(s):  
Peter Keevash
Keyword(s):  
Turan Problems

scholarly journals Rainbow Turán Problems for Paths and Forests of Stars

10.37236/6430 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Daniel Johnston ◽  
Cory Palmer ◽  
Amites Sarkar
Keyword(s):  
Systematic Study ◽  
Colored Graph ◽  
Turán Number ◽  
Turan Problems ◽  
Different Color

For a fixed graph $F$, we would like to determine the maximum number of edges in a properly edge-colored graph on $n$ vertices which does not contain a rainbow copy of $F$, that is, a copy of $F$ all of whose edges receive a different color. This maximum, denoted by $ex^*(n,F)$, is the rainbow Turán number of $F$, and its systematic study was initiated by Keevash, Mubayi, Sudakov and Verstraëte [Combinatorics, Probability and Computing 16 (2007)]. We determine $ex^*(n,F)$ exactly when $F$ is a forest of stars, and give bounds on $ex^*(n,F)$ when $F$ is a path with $l$ edges, disproving a conjecture in the aforementioned paper for $l=4$.

scholarly journals Chromatic Turán problems and a new upper bound for the Turán density of $\mathcal{K}_4^-$

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
John Talbot
Keyword(s):  
Upper Bound ◽  
Turan Problems ◽  
International Audience ◽  
New Type ◽  
Extremal Hypergraph

International audience We consider a new type of extremal hypergraph problem: given an $r$-graph $\mathcal{F}$ and an integer $k≥2$ determine the maximum number of edges in an $\mathcal{F}$-free, $k$-colourable $r$-graph on $n$ vertices. Our motivation for studying such problems is that it allows us to give a new upper bound for an old problem due to Turán. We show that a 3-graph in which any four vertices span at most two edges has density less than $\frac{33}{ 100}$, improving previous bounds of $\frac{1}{ 3}$ due to de Caen [1], and $\frac{1}{ 3}-4.5305×10^-6$ due to Mubayi [9].

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