Related papers: Degenerate Tur\'an densities of sparse hypergraphs

We consider the problem of estimating the edge density of densest $K$-node subgraphs of an Erd\"os-R\'{e}nyi graph $\mathbb{G}(n,1/2)$. The problem is well-understood in the regime $K=\Theta(\log n)$ and in the regime $K=\Theta(n)$. In the…

We study the generalized Ramsey--Tur\'an function $\mathrm{RT}(n,K_s,K_t,o(n))$, which is the maximum possible number of copies of $K_s$ in an $n$-vertex $K_t$-free graph with independence number $o(n)$. The case when $s=2$ was settled by…

Let $P$ denote a 3-uniform hypergraph consisting of 7 vertices $a,b,c,d,e,f,g$ and 3 edges $\{a,b,c\}, \{c,d,e\},$ and $\{e,f,g\}$. It is known that the $r$-color Ramsey number for $P$ is $R(P;r)=r+6$ for $r\le 7$. The proof of this result…

Confirming a conjecture of Vera T. S\'os in a very strong sense, we give a complete solution to Tur\'an's hypergraph problem for the Fano plane. That is we prove for $n\ge 8$ that among all $3$-uniform hypergraphs on $n$ vertices not…

Given integers $r > t \ge 1$ and a real number $p > 0$, the $(t,p)$-norm $\left\lVert \mathcal{H} \right\rVert_{t,p}$ of an $r$-graph $\mathcal{H}$ is the sum of the $p$-th power of the degrees $d_{\mathcal{H}}(T)$ over all $t$-subsets $T…

An ordered graph $H$ is a simple graph with a linear order on its vertex set. The corresponding Tur\'an problem, first studied by Pach and Tardos, asks for the maximum number $\text{ex}_<(n,H)$ of edges in an ordered graph on $n$ vertices…

A {\it simple $k$-coloring} of a multigraph $G$ is a decomposition of the edge multiset as a disjoint sum of $k$ simple graphs which are referred as colors. A subgraph $H$ of a multigraph $G$ is called {\it multicolored} if its edges…

We prove that, for any finite set of minimal $r$-graph patterns, there is a finite family $\mathcal F$ of forbidden $r$-graphs such that the extremal Tur\'an constructions for $\mathcal F$ are precisely the maximum $r$-graphs obtainable…

The analysis of large simple graphs with extreme values of the densities of edges and triangles has been extended to the statistical structure of typical graphs of fixed intermediate densities, by the use of large deviations of Erdoes-Renyi…

The \emph{Ramsey multiplicity constant} of a graph $H$ is the limit as $n$ tends to infinity of the minimum density of monochromatic labeled copies of $H$ in a $2$-edge colouring of $K_n$. Fox and Wigderson recently identified a large…

Given a graph $H$ and a function $f:\mathbb{Z}^+ \longrightarrow \mathbb{Z}^+ $, the Ramsey-Tur\'an number of $H$ and $f$, denoted by $RT(n, H, f(n))$, is the maximum number of edges a graph $G$ on $n$ vertices can have, which does not…

Let $F$ be a graph. We say that a hypergraph $H$ is a {\it Berge}-$F$ if there is a bijection $f : E(F) \rightarrow E(H )$ such that $e \subseteq f(e)$ for every $e \in E(F)$. Note that Berge-$F$ actually denotes a class of hypergraphs. The…

Let $\cal H$ be a family of graphs. The Tur\'an number ${\rm ex}(n,{\cal H})$ is the maximum possible number of edges in an $n$-vertex graph which does not contain any member of $\cal H$ as a subgraph. As a common generalization of…

Given a family of $k$-hypergraphs $\mathcal{F}$, $ex(n,\mathcal{F})$ is the maximum number of edges a $k$-hypergraph can have, knowing that said hypergraph has $n$ vertices but contains no copy of any hypergraph from $\mathcal{F}$ as a…

For a nondegenerate $r$-graph $F$, large $n$, and $t$ in the regime $[0, c_{F} n]$, where $c_F>0$ is a constant depending only on $F$, we present a general approach for determining the maximum number of edges in an $n$-vertex $r$-graph that…

For positive integers $n \ge s > r$, a \emph{Tur\'{a}n $(n,s,r)$-system} is an $n$-vertex $r$-graph in which every set of $s$ vertices contains at least one edge. Let $T(n,s,r)$ denote the the minimum size of a Tur\'{a}n $(n,s,r)$-system.…

The \emph{minimum positive co-degree} of a non-empty $r$-graph ${H}$, denoted $\delta_{r-1}^+( {H})$, is the maximum $k$ such that if $S$ is an $(r-1)$-set contained in a hyperedge of $ {H}$, then $S$ is contained in at least $k$ distinct…

Recently, several hypergraph Tur\'{a}n problems were solved by the powerful random algebraic method. However, the random algebraic method usually requires some parameters to be very large, hence we are concerned about how these Tur\'{a}n…

A classical theorem of Ahlswede and Katona determines the maximum density of the $2$-edge star in a graph with a given edge density. Motivated by its application in hypergraph Tur\'{a}n problems, we establish a refinement of their result…

A $k$-path is a hypergraph P_k = e_1,e_2,...,e_k such that |e_i \cap e_j| = 1 if |j - i| = 1 and e_i \cap e_j is empty otherwise. A k-cycle is a hypergraph C_k = e_1,e_2,.. ,e_k obtained from a (k-1)-path e_1,e_2,...,e_{k-1} by adding an…