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In this paper, we introduce a class of graphs which we call average hereditary graphs. Many graphs that occur in the usual graph theory applications belong to this class of graphs. Many popular types of graphs fall under this class, such as…
We consider the problem of decomposing some $t$-uniform hypergraph $G$ into copies of another, say $H$, with nonnegative rational weights. For fixed $H$ on $k$ vertices, we show that this is always possible for all $G$ having sufficiently…
A graph homomorphism between two graphs is a map from the vertex set of one graph to the vertex set of the other graph, that maps edges to edges. In this note we study the range of a uniformly chosen homomorphism from a graph G to the…
For two graphs $F$ and $H$, the relative Tur\'{a}n number $\mathrm{ex}(H,F)$ is the maximum number of edges in an $F$-free subgraph of $H$. Foucaud, Krivelevich, and Perarnau \cite{FKP} and Perarnau and Reed \cite{PR} studied these…
One of the fundamental results in graph minor theory is that for every planar graph $H$, there is a minimum integer $f(H)$ such that graphs with no minor isomorphic to $H$ have treewidth at most $f(H)$. A lower bound for ${f(H)}$ can be…
An edge-colored graph is said to be rainbow if all its edges have distinct colors. In this paper, we study the rainbow analogue of a fundamental result of Mader [\emph{Math. Ann.} \textbf{174} (1967), 265--268] on the existence of…
Finding dense subgraphs of a large graph is a standard problem in graph mining that has been studied extensively both for its theoretical richness and its many practical applications. In this paper we introduce a new family of dense…
We establish new lower bounds for the Tur\'an and Zarankiewicz numbers of certain apex partite hypergraphs. Given a $(d-1)$-partite $(d-1)$-uniform hypergraph $\mathcal{H}$, let $\mathcal{H}(k)$ be the $d$-partite $d$-uniform hypergraph…
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…
We prove tight bounds on the site percolation threshold for $k$-uniform hypergraphs of maximum degree $\Delta$ and for $k$-uniform hypergraphs of maximum degree $\Delta$ in which any pair of edges overlaps in at most $r$ vertices. The…
In 1975, Erd\H{o}s and Sauer asked to estimate, for any constant $r$, the maximum number of edges an $n$-vertex graph can have without containing an $r$-regular subgraph. In a recent breakthrough, Janzer and Sudakov proved that any…
The planted densest subgraph detection problem refers to the task of testing whether in a given (random) graph there is a subgraph that is unusually dense. Specifically, we observe an undirected and unweighted graph on $n$ vertices. Under…
A sequence of nonnegative integers \pi =(d_1,d_2,...,d_n) is graphic if there is a (simple) graph G with degree sequence \pi. In this case, G is said to realize or be a realization of \pi. Degree sequence results in the literature generally…
We improve the best known upper bound on the number of edges in a unit-distance graph on $n$ vertices for each $n\in\{16,\ldots,30\}$. When $n\leq 21$, our bounds match the best known lower bounds, and we fully enumerate the densest…
An (edge) decomposition of a graph $G$ is a set of subgraphs of $G$ whose edge sets partition the edge set of $G$. Here we show, for each odd $\ell \geq 5$, that any graph $G$ of sufficiently large order $n$ with minimum degree at least…
A well-known theorem of R\"odl says that for every graph $H$, and every $\epsilon>0$, there exists $\delta>0$ such that if $G$ does not contain an induced copy of $H$, then there exists $X\subseteq V(G)$ with $|X|\ge \delta|G|$ such that…
Given an $r$-graph $F$ with $r \ge 2$, let $\mathrm{ex}(n, (t+1) F)$ denote the maximum number of edges in an $n$-vertex $r$-graph with at most $t$ pairwise vertex-disjoint copies of $F$. Extending several old results and complementing…
Dense subgraph discovery is an important graph-mining primitive with a variety of real-world applications. One of the most well-studied optimization problems for dense subgraph discovery is the densest subgraph problem, where given an…
We consider the structure of $H$-free subgraphs of graphs with high minimal degree. We prove that for every $k>m$ there exists an $\epsilon:=\epsilon(k,m)>0$ so that the following holds. For every graph $H$ with chromatic number $k$ from…
Consider a host hypergraph $G$ which contains a spanning structure due to minimum degree considerations. We collect three results proving that if the edges of $G$ are sampled at the appropriate rate then the spanning structure still appears…