Related papers: Localized Version of Hypergraph Erdos-Gallai Theor…

A good edge-labelling of a simple graph is a labelling of its edges with real numbers such that, for any ordered pair of vertices (u,v), there is at most one nondecreasing path from u to v. Say a graph is good if it admits a good…

The Tur\'an number $\text{ex}(n,H)$ of a graph $H$ is the maximal number of edges in an $H$-free graph on $n$ vertices. In $1983$ Chung and Erd\H{o}s asked which graphs $H$ with $e$ edges minimize $\text{ex}(n,H)$. They resolved this…

For an integer $r\geqslant 3$, a hypergraph on vertex set $[n]$ is $r$-uniform if each edge is a set of $r$ vertices, and is said to be linear if every two distinct edges share at most one vertex. Given a family $\mathcal{H}$ of linear…

We obtain some new upper bounds on the maximum number $f(n)$ of edges in $n$-vertex graphs without containing cycles of length four. This leads to an asymptotically optimal bound on $f(n)$ for a broad range of integers $n$ as well as a…

A classic theorem of Erd\H{o}s and P\'osa (1965) states that every graph has either $k$ vertex-disjoint cycles or a set of $O(k \log k)$ vertices meeting all its cycles. While the standard proof revolves around finding a large `frame' in…

Given a hypergraph $H=(V,E)$, define for every edge $e\in E$ a linear expression with arguments corresponding with the vertices. Next, let the polynomial $p_H$ be the product of such linear expressions for all edges. Our main goal was to…

Learning a hidden hypergraph is a natural generalization of the classical group testing problem that consists in detecting unknown hypergraph $H_{un}=H(V,E)$ by carrying out edge-detecting tests. In the given paper we focus our attention…

The conjecture of Brown, Erd\H{o}s and S\'os from 1973 states that, for any $k \ge 3$, if a $3$-uniform hypergraph $H$ with $n$ vertices does not contain a set of $k+3$ vertices spanning at least $k$ edges then it has $o(n^2)$ edges. The…

For a $k$-uniform hypergraph $F$ let $\textrm{ex}(n,F)$ be the maximum number of edges of a $k$-uniform $n$-vertex hypergraph $H$ which contains no copy of $F$. Determining or estimating $\textrm{ex}(n,F)$ is a classical and central problem…

The Erd\H{o}s-Simonovits stability theorem is one of the most widely used theorems in extremal graph theory. We obtain an Erd\H{o}s-Simonovits type stability theorem in multi-partite graphs. Different from the Erd\H{o}s-Simonovits stability…

In 1975, P. Erd\H{o}s proposed the problem of determining the maximum number $f(n)$ of edges in a graph on $n$ vertices in which any two cycles are of different lengths. Let $f^{\ast}(n)$ be the maximum number of edges in a simple graph on…

In this paper, we study the maximum number of edges in an $N$-vertex $r$-uniform hypergraph with girth $g$ where $g \in \{5,6 \}$. Writing $\textrm{ex}_r ( N, \mathcal{C}_{<g} )$ for this maximum, it is shown that $\textrm{ex}_r ( N ,…

A Berge cycle of length $\ell$ in a hypergraph is an alternating sequence of $\ell$ distinct vertices and $\ell$ distinct edges $v_1,e_1,v_2, \ldots, v_\ell, e_{\ell}$ such that $\{v_i, v_{i+1}\} \subseteq e_i$ for all $i$, with indices…

For a fixed graph $F$, let $ex_F(G)$ denote the size of the largest $F$-free subgraph of $G$. Computing or estimating $ex_F(G)$ for various pairs $F,G$ is one of the central problems in extremal combinatorics. It is thus natural to ask how…

We establish the following strengthening of the celebrated Andr{\'a}sfai--Erd\H{o}s--S\'{o}s theorem: If $G$ is an $n$-vertex $K_{r+1}$-free graph whose minimum degree $\delta(G)$ and maximum degree $\Delta(G)$ satisfy \begin{align*}…

A k-edge-weighting of a graph G is a function w: E(G)->{1,2,...,k}. An edge-weighting naturally induces a vertex coloring c, where for every vertex v in V(G), c(v) is sum of weights of the edges that are adjacent to vertex v. If the induced…

The Lagrangian density of an $r$-uniform hypergraph $H$ is $r!$ multiplying the supremum of the Lagrangians of all $H$-free $r$-uniform hypergraphs. For an $r$-uniform graph $H$ with $t$ vertices, it is clear that $\pi_{\lambda}(H)\ge…

Haj\'os conjectured in 1968 that every Eulerian \(n\)-vertex graph can be decomposed into at most $\lfloor (n-1)/2\rfloor$ edge-disjoint cycles. This has been confirmed for some special graph classes, but the general case remains open. In a…

For $r\geq 2$ and $p\geq 1$, the $p$-spectral radius of an $r$-uniform hypergraph $H=(V,E)$ on $n$ vertices is defined to be $$\rho_p(H)=\max_{{\bf x}\in \mathbb{R}^n: \|{\bf x}\|_p=1}r \cdot \!\!\!\! \sum_{\{i_1,i_2,\ldots, i_r\}\in E(H)}…

By a tight tour in a $k$-uniform hypergraph $H$ we mean any sequence of its vertices $(w_0,w_1,\ldots,w_{s-1})$ such that for all $i=0,\ldots,s-1$ the set $e_i=\{w_i,w_{i+1}\ldots,w_{i+k-1}\}$ is an edge of $H$ (where operations on indices…