Related papers: A problem of Erdos and Sos on 3-graphs

A $k$-book in a hypergraph consists of $k$ Berge triangles sharing a common edge. In this paper we prove that the number of the hyperedges in a $k$-book-free 3-uniform hypergraph on $n$ vertices is at most $\frac{n^2}{8}(1+o(1))$.

Let G=(V,E) be a graph. Let k < |V| be an integer. Let O_k be the number of edge induced subgraphs of G having k vertices and an odd number of edges. Let E_k be the number of edge induced subgraphs of G having k vertices and an even number…

In this paper we raise a variant of a classic problem in extremal graph theory, which is motivated by a design of fractional repetition codes, a model in distributed storage systems. For any feasible positive integers $d\geq 3$, $n \geq 3$,…

A perfect matching in a 4-uniform hypergraph is a subset of $\lfloor\frac{n}{4}\rfloor$ disjoint edges. We prove that if $H$ is a sufficiently large 4-uniform hypergraph on $n=4k$ vertices such that every vertex belongs to more than…

A famous conjecture of Erd\H{o}s and S\'os states that every graph with average degree more than $k - 1$ contains all trees with $k$ edges as subgraphs. We prove that the Erd\H{o}s-S\'os conjecture holds approximately, if the size of the…

Let $G$ be a $K_4$-free graph, an edge in its complement is a $K_4$-\emph{saturating} edge if the addition of this edge to $G$ creates a copy of $K_4$. Erd\H{o}s and Tuza conjectured that for any $n$-vertex $K_4$-free graph $G$ with…

An $r$-cut of a $k$-uniform hypergraph is a partition of its vertex set into $r$ parts, and the size of the cut is the number of edges which have at least one vertex in each part. The study of the possible size of the largest $r$-cut in a…

A connected graph G is 3-flow-critical if G does not have a nowhere-zero 3-flow, but every proper contraction of G does. We prove that every n-vertex 3-flow-critical graph other than K_2 and K_4 has at least 5n/3 edges. This bound is tight…

A topological graph is $k$-quasi-planar if it does not contain $k$ pairwise crossing edges. A 20-year-old conjecture asserts that for every fixed $k$, the maximum number of edges in a $k$-quasi-planar graph on $n$ vertices is $O(n)$. Fox…

We investigate extremal problems for quasirandom hypergraphs. We say that a $3$-uniform hypergraph $H=(V,E)$ is $(d,\eta)$-quasirandom if for any subset $X\subseteq V$ and every set of pairs $P\subseteq V\times V$ the number of pairs…

Given a set S of n points in R^D, and an integer k such that 0 <= k < n, we show that a geometric graph with vertex set S, at most n - 1 + k edges, maximum degree five, and dilation O(n / (k+1)) can be computed in time O(n log n). For any…

We show that a 2-subset-regular self-complementary 3-uniform hypergraph with $n$ vertices exists if and only if $n\ge 6$ and $n$ is congruent to 2 modulo 4.

A set $R\subseteq E(G)$ of a graph $G$ is $k$-removable if $G-R$ has a nowhere-zero $k$-flow. We prove that every graph $G$ admitting a nowhere-zero $4$-flow has a $3$-removable subset consisting of at most $\frac{1}{6}|E(G)|$ edges. This…

The classical Andr\'{a}sfai--Erd\H{o}s--S\'{o}s Theorem states that for $\ell\ge 2$, every $n$-vertex $K_{\ell+1}$-free graph with minimum degree greater than $\frac{3\ell-4}{3\ell-1}n$ must be $\ell$-partite. We establish a simple…

For $n\geq 3$ and $r=r(n) \geq 3$, let $\boldsymbol{k} =\boldsymbol{k}(n)=(k_1, \ldots, k_n)$ be a sequence of non-negative integers with sum $M(\boldsymbol{k})=\sum_{j=1}^{n} k_j$. We assume that $M(\boldsymbol{k})$ is divisible by $r$ for…

In this paper we study the fundamental problem of finding small dense subgraphs in a given graph. For a real number $s>2$, we prove that every graph on $n$ vertices with average degree at least $d$ contains a subgraph of average degree at…

In this note, we prove several Tur\'an-type results on geometric hypergraphs. The two main theorems are 1) Every $n$-vertex geometric 3-hypergraph in 2-space with no three strongly crossing edges has at most $O(n^2)$ edges, 2) Every…

For a real c \geq 1 and an integer n, let f(n,c) denote the maximum integer f so that every graph on n vertices contains an induced subgraph on at least f vertices in which the maximum degree is at most c times the minimum degree. Thus, in…

We study the maximum number of hyperedges in a 3-uniform hypergraph on $n$ vertices that does not contain a Berge cycle of a given length $\ell$. In particular we prove that the upper bound for $C_{2k+1}$-free hypergraphs is of the order…

Let $G$ and $H$ be $k$-graphs ($k$-uniform hypergraphs); then a perfect $H$-packing in $G$ is a collection of vertex-disjoint copies of $H$ in $G$ which together cover every vertex of $G$. For any fixed $H$ let $\delta(H, n)$ be the minimum…