scholarly journals Vertex Turán problems for the oriented hypercube

2021 ◽  
Vol 13 (2) ◽  
pp. 356-366
Author(s):  
Dániel Gerbner ◽  
Abhishek Methuku ◽  
Dániel T. Nagy ◽  
Balázs Patkós ◽  
Máté Vizer
Keyword(s):  
Short Note ◽  
Oriented Graph ◽  
Directed Path ◽  
Maximum Cardinality ◽  
Directed Tree ◽  
Induced Subgraph ◽  
Asymptotic Value ◽  
Turan Problem ◽  
Turan Problems ◽  
Turán Problem

Abstract In this short note we consider the oriented vertex Turán problem in the hypercube: for a fixed oriented graph F → \vec F , determine the maximum cardinality e x v ( F → , Q → n ) e{x_v}\left( {\vec F,{{\vec Q}_n}} \right) of a subset U of the vertices of the oriented hypercube Q → n {\vec Q_n} such that the induced subgraph Q → n [ U ] {\vec Q_n}\left[ U \right] does not contain any copy of F → \vec F . We obtain the exact value of e x v ( P k , →   Q n → ) e{x_v}\left( {\overrightarrow {{P_k},} \,\overrightarrow {{Q_n}} } \right) for the directed path P k → \overrightarrow {{P_k}} , the exact value of e x v ( V 2 → ,   Q n → ) e{x_v}\left( {\overrightarrow {{V_2}} ,\,\overrightarrow {{Q_n}} } \right) for the directed cherry V 2 → \overrightarrow {{V_2}} and the asymptotic value of e x v ( T → , Q n → ) e{x_v}\left( {\overrightarrow T ,\overrightarrow {{Q_n}} } \right) for any directed tree T → \vec T .

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.

scholarly journals Generalizations of the Ruzsa–Szemerédi and rainbow Turán problems for cliques

2020 ◽  
pp. 1-18
Author(s):  
W. T. Gowers ◽  
Barnabás Janzer
Keyword(s):  
Natural Generalization ◽  
Turan Problem ◽  
Turan Problems ◽  
Turán Problem ◽  
Positive Integers

Abstract Considering a natural generalization of the Ruzsa–Szemerédi problem, we prove that for any fixed positive integers r, s with r < s, there are graphs on n vertices containing $n^{r}e^{-O\left(\sqrt{\log{n}}\right)}=n^{r-o(1)}$ copies of K s such that any K r is contained in at most one K s . We also give bounds for the generalized rainbow Turán problem ex (n, H, rainbow - F) when F is complete. In particular, we answer a question of Gerbner, Mészáros, Methuku and Palmer, showing that there are properly edge-coloured graphs on n vertices with $n^{r-1-o(1)}$ copies of K r such that no K r is rainbow.

Inverting the Turán problem with chromatic number

2021 ◽  
Vol 344 (9) ◽  
pp. 112517
Author(s):  
Xiutao Zhu ◽  
Yaojun Chen
Keyword(s):  
Chromatic Number ◽  
Turan Problem ◽  
Turán Problem

scholarly journals On a Turán problem in weakly quasirandom 3-uniform hypergraphs

2018 ◽  
Vol 20 (5) ◽  
pp. 1139-1159 ◽  
Author(s):  
Christian Reiher ◽  
Vojtěch Rödl ◽  
Mathias Schacht
Keyword(s):  
Uniform Hypergraphs ◽  
Turan Problem ◽  
Turán Problem

scholarly journals An Ordered Turán Problem for Bipartite Graphs

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.

scholarly journals Unified approach to the generalized Turán problem and supersaturation

2022 ◽  
Vol 345 (3) ◽  
pp. 112743
Author(s):  
Dániel Gerbner ◽  
Zoltán Lóránt Nagy ◽  
Máté Vizer
Keyword(s):  
Unified Approach ◽  
Turan Problem ◽  
Turán Problem

Exact solution of the hypergraph Turán problem for k-uniform linear paths

COMBINATORICA ◽  
2014 ◽  
Vol 34 (3) ◽  
pp. 299-322 ◽  
Author(s):  
Zoltán Füredi ◽  
Tao Jiang ◽  
Robert Seiver
Keyword(s):  
Exact Solution ◽  
Turan Problem ◽  
Turán Problem

scholarly journals The Size of a Hypergraph and its Matching Number

2012 ◽  
Vol 21 (3) ◽  
pp. 442-450 ◽  
Author(s):  
HAO HUANG ◽  
PO-SHEN LOH ◽  
BENNY SUDAKOV
Keyword(s):  
Matching Number ◽  
Uniform Hypergraph ◽  
Edge Matching ◽  
Turan Problem ◽  
Turán Problem

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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