Let $H$ be a graph on $n$ vertices and let the blow-up graph
$G[H]$ be defined as follows. We replace each vertex $v_i$ of $H$ by a cluster
$A_i$ and connect some pairs of vertices of $A_i$ and $A_j$ if $(v_i,v_j)$ was
an edge of the graph $H$. As usual, we define the edge density between $A_i$ and $A_j$ as $d(A_i,A_j)=\frac{e(A_i,A_j)}{|A_i||A_j|}.$ We study the following problem. Given densities $γ_{ij}$ for each edge $(i,j)\in E(H)$. Then one has to decide whether there exists a blow-up graph $G[H]$ with edge densities at least $γ_{ij}$ such that one cannot choose a vertex from each cluster so that the obtained graph is isomorphic to $H$, i.e, no $H$ appears as a transversal in $G[H]$. We call $d_{crit}(H)$ the maximal value for which there exists a blow-up graph $G[H]$ with edge densities $d(A_i,A_j)=d_{crit}(H)$ $((v_i,v_j)\in E(H))$ not containing $H$ in the above sense. Our main goal is to determine the critical edge density and to characterize the extremal graphs.
Reiher, Rödl and Schacht [J. London Math. Soc. 97 (2018), 77--97] showed that the uniform Turán density of every $3$-uniform hypergraph is either $0$ or at least $1/27$, and asked whether there exist $3$-uniform hypergraphs with uniform Turán density equal or arbitrarily close to $1/27$. We construct $3$-uniform hypergraphs with uniform Turán density equal to $1/27$.
In the early 1980s, Erdős and Sós initiated the study of the classical Turán problem with a uniformity condition: the uniform Turán density of a hypergraph $H$ is the infimum over all $d$ for which any sufficiently large hypergraph with the property that all its linear-size subhyperghraphs have density at least $d$ contains $H$. In particular, they raise the questions of determining the uniform Turán densities of $K_4^{(3)-}$ and $K_4^{(3)}$. The former question was solved only recently in [Israel J. Math. 211 (2016), 349-366] and [J. Eur. Math. Soc. 20 (2018), 1139-1159], while the latter sti…
Let $\mathcal{H}$ be an $r$-uniform hypergraph. The Turán number $\text{ex}(n,\mathcal{H})$ is the maximum number of edges in an $n$-vertex $\mathcal{H}$-free $r$-uniform hypergraph. The Turán density of $\mathcal{H}$ is defined by \[π(\mathcal{H})=\lim_{n\rightarrow\infty}\frac{\text{ex}(n,\mathcal{H})}{\binom{n}{r}}.\] In this paper, we consider the Turán density of projective geometries. We give two new constructions of $PG_{m}(q)$-free hypergraphs which improve some results given by Keevash (J. Combin. Theory Ser. A, 111: 289--309, 2005). Based on an upper bound of blocking sets of $PG_m(q…
We consider the Turán problems of $2$-edge-colored graphs. A $2$-edge-colored graph $H=(V, E_r, E_b)$ is a triple consisting of the vertex set $V$, the set of red edges $E_r$ and the set of blue edges $E_b$ with $E_r$ and $E_b$ do not have to be disjoint. The Turán density $π(H)$ of $H$ is defined to be $\lim_{n\to\infty} \max_{G_n}h_n(G_n)$, where $G_n$ is chosen among all possible $2$-edge-colored graphs on $n$ vertices containing no $H$ as a sub-graph and $h_n(G_n)=(|E_r(G)|+|E_b(G)|)/{n\choose 2}$ is the formula to measure the edge density of $G_n$. We will determine the Turán densities of…
Given $α>0$ and an integer $\ell\geq5$, we prove that every sufficiently large $3$-uniform hypergraph $H$ on $n$ vertices in which every two vertices are contained in at least $αn$ edges contains a copy of $C_\ell^{-}$, a tight cycle on $\ell$ vertices minus one edge. This improves a previous result by Balogh, Clemen, and Lidický.
In this paper, we consider the Turán problems on $\{1,3\}$-hypergraphs. We prove that a $\{1, 3\}$-hypergraph is degenerate if and only if it's $H^{\{1, 3\}}_5$-colorable, where $H^{\{1, 3\}}_5$ is a hypergraph with vertex set $V=[5]$ and edge set $E=\{\{2\}, \{3\}, \{1, 2, 4\}, \{1, 3, 5\}, \{1, 4, 5\}\}.$ Using this result, we further prove that for any finite set $R$ of distinct positive integers, except the case $R=\{1, 2\}$, there always exist non-trivial degenerate $R$-graphs. We also compute the Turán densities of some small $\{1,3\}$-hypergraphs.
Hypergraphs with uniform Turán density equal to 8/27
Institute of Computer Science, Czech Academy of Sciences（捷克科学院计算机研究所）；Universität Heidelberg（海德堡大学）；Faculty of Informatics, Masaryk University（马萨里克大学信息学院）；Leipzig University（莱比锡大学）；University of Warwick（华威大学）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学促进会数学研究所）；Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS)（基础科学研究院极值组合与概率组）
P. Erdős [On extremal problems of graphs and generalized graphs, Israel Journal of Mathematics 2 (1964), 183-190] characterised those hypergraphs $F$ that have to appear in any sufficiently large hypergraph $H$ of positive density. We study related questions for $3$-uniform hypergraphs with the additional assumption that $H$ has to be uniformly dense with respect to vertex sets. In particular, we characterise those hypergraphs $F$ that are guaranteed to appear in large uniformly dense hypergraphs $H$ of positive density. We also review the case when the density of the induced subhypergraphs of…
Given a $k$-graph $H$ a complete blow-up of $H$ is a $k$-graph $\hat{H}$ formed by replacing each $v\in V(H)$ by a non-empty vertex class $A_v$ and then inserting all edges between any $k$ vertex classes corresponding to an edge of $H$. Given a subgraph $G\subseteq \hat{H}$ and an edge $e\in E(H)$ we define the density $d_e(G)$ to be the proportion of edges present in $G$ between the classes corresponding to $e$. The density Turán problem for $H$ asks: determine the minimal value $d_{crit}(H)$ such that any subgraph $G\subseteq \hat{H}$ satisfying $d_e(G)> d_{crit}(H)$ for every $e\in E(H)$ co…
Let $G$ be a finite abelian group, and $r$ be a multiple of its exponent. The generalized Erdős-Ginzburg-Ziv constant $s_r(G)$ is the smallest integer $s$ such that every sequence of length $s$ over $G$ has a zero-sum subsequence of length $r$. We show that $s_{2m}(\mathbb{Z}_2^d) \leq C_m 2^{d/m} + O(1)$ when $d\rightarrow\infty$, and $s_{2m}(\mathbb{Z}_2^d) \geq 2^{d/m} + 2m-1$ when $d=km$. We use results on $s_r(G)$ to prove new bounds for the codegree Turán density of complete $r$-graphs.
