Related papers: On the decomposition of random hypergraphs

Let $H$ be a fixed graph. A {\em fractional $H$-decomposition} of a graph $G$ is an assignment of nonnegative real weights to the copies of $H$ in $G$ such that for each $e \in E(G)$, the sum of the weights of copies of $H$ containing $e$…

A famous conjecture (usually called Ryser's conjecture) that appeared in the Ph.D thesis of his student, J.~R.~Henderson [15], states that for an $r$-uniform $r$-partite hypergraph $\mathcal{H}$, the inequality…

Given a sufficiently large and sufficiently dense bipartite graph $G=(A, B; E),$ we present a novel method for decomposing the majority of the edges of $G$ into quasirandom graphs so that the vertex sets of these quasirandom graphs…

A $k$-uniform hypergraph is $s$-almost intersecting if every edge is disjoint from exactly $s$ other edges. Gerbner, Lemons, Palmer, Patk\'os and Sz\'ecsi conjectured that for every $k$, and $s>s_0(k)$, every $k$-uniform $s$-almost…

Consider a random graph G in G(n,p) and the graph property: G contains a copy of a specific graph H. (Note: H depends on n; a motivating example: H is a Hamiltonian cycle.) Let q be the minimal value for which the expected number of copies…

For a given graph $F$, the $F$-saturation number of a graph $G$, denoted by $ {sat}(G, F)$, is the minimum number of edges in an edge-maximal $F$-free subgraph of $G$. In 2017, Kor\'andi and Sudakov determined $ {sat}({G}(n, p), K_r)$…

The independence number $\alpha(H)$ of a hypergraph $H$ is the maximum cardinality of a set of vertices of $H$ that does not contain an edge of $H$. Generalizing Shearer's classical lower bound on the independence number of triangle-free…

Let $H_{n,(p_m)_{m=2,\ldots,M}}$ be a random non-uniform hypergraph of dimension $M$ on $2n$ vertices, where the vertices are split into two disjoint sets of size $n$, and colored by two distinct colors. Each non-monochromatic edge of size…

Let $\mu(G)$ denote the minimum number of edges whose addition to $G$ results in a Hamiltonian graph, and let $\hat{\mu}(G)$ denote the minimum number of edges whose addition to $G$ results in a pancyclic graph. We study the distributions…

By using the Szemer\'edi Regularity Lemma, Alon and Sudakov recently extended the classical Andr\'asfai-Erd\~os-S\'os theorem to cover general graphs. We prove, without using the Regularity Lemma, that the following stronger statement is…

Let $\mathcal{H}$ be an $r$-uniform hypergraph and $F$ be a graph. We say $\mathcal{H}$ contains $F$ as a trace if there exists some set $S\subseteq V(\mathcal{H})$ such that $\mathcal{H}|_{S}:=\{E\cap S: E\in E(\mathcal{H})\}$ contains a…

In the model of randomly perturbed graphs we consider the union of a deterministic graph $\mathcal{G}_\alpha$ with minimum degree $\alpha n$ and the binomial random graph $\mathbb{G}(n,p)$. This model was introduced by Bohman, Frieze, and…

For integers $r,t\geq2$ and $n\geq1$ let $f_r(t,n)$ be the minimum, over all factorizations of the complete $r$-uniform hypergraph of order $n$ into $t$ factors $H_1,\dots,H_t$, of $\sum_{i=1}^tc(H_i)$ where $c(H_i)$ is the number 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 $H$.…

Let $H_k^r$ denote an $r$-uniform hypergraph with $k$ edges and $r+1$ vertices, where $k \leq r+1$ (it is easy to see that such a hypergraph is unique up to isomorphism). The known general bounds on its Tur\'{a}n density are $\pi(H_k^r)…

Let $F=\{H_1,...,H_k\}$ be a family of graphs. A graph $G$ with $m$ edges is called {\em totally $F$-decomposable} if for {\em every} linear combination of the form $\alpha_1 e(H_1) + ... + \alpha_k e(H_k) = m$ where each $\alpha_i$ is a…

Let $r\ge 3$. Given an $r$-graph $H$, the minimum codegree $\delta_{r-1}(H)$ is the largest integer $t$ such that every $(r-1)$-subset of $V(H)$ is contained in at least $t$ edges of $H$. Given an $r$-graph $F$, the codegree Tur\'an density…

We prove that for integers $2 \leq \ell < k$ and a small constant $c$, if a $k$-uniform hypergraph with linear minimum codegree is randomly `perturbed' by changing non-edges to edges independently at random with probability $p \geq…

For a graph $G$ and a hereditary property $\mathcal{P}$, let $\text{ex}(G,\mathcal{P})$ denote the maximum number of edges of a subgraph of $G$ that belongs to $\mathcal{P}$. We prove that for every non-trivial hereditary property…

Consider integers $k,\ell$ such that $0\le \ell \le \binom{k}2$. Given a large graph $G$, what is the fraction of $k$-vertex subsets of $G$ which span exactly $\ell$ edges? When $G$ is empty or complete, and $\ell$ is zero or…