Related papers: Best possible bounds on the double-diversity of in…

For a (possibly infinite) fixed family of graphs F, we say that a graph G overlays F on a hypergraph H if V(H) is equal to V(G) and the subgraph of G induced by every hyperedge of H contains some member of F as a spanning subgraph.While it…

A perfect matching of a complete graph $K_{2n}$ is a 1-regular subgraph that contains all the vertices. Two perfect matchings intersect if they share an edge. It is known that if $\mathcal{F}$ is family of intersecting perfect matchings of…

The well-known Erd\H{o}s--Ko--Rado theorem states that for $n> 2k$, every intersecting family of $k$-sets of $[n]:=\{1,\ldots ,n\}$ has at most $ {n-1 \choose k-1}$ sets, and the extremal family consists of all $k$-sets containing a fixed…

Given a 2-edge-coloring $f : E(K_n) \rightarrow \{\pm 1\}$, the discrepancy of a subgraph $F \subseteq K_n$ is defined as $\left| \sum_{e \in E(F)} f(e) \right|$. Erd\H{o}s, F\"uredi, Loebl and S\'os showed that if $F$ is an $n$-vertex tree…

In this short note, we address two problems in extremal set theory regarding intersecting families. The first problem is a question posed by Kupavskii: is it true that given two disjoint cross-intersecting families $\mathcal{A}, \mathcal{B}…

Let $G=(V,E)$ be a graph of density $p$ on $n$ vertices. Following Erd\H{o}s, \L uczak and Spencer, an $m$-vertex subgraph $H$ of $G$ is called {\em full} if $H$ has minimum degree at least $p(m - 1)$. Let $f(G)$ denote the order of a…

Two families $\mathcal{F}$ and $\mathcal{G}$ are called cross-intersecting if for every $F\in \mathcal{F}$ and $G\in \mathcal{G}$, the intersection $F\cap G$ is non-empty. It is significant to determine the maximum sum of sizes of…

For an $n$-element set $X$ let $\binom{X}{k}$ be the collection of all its $k$-subsets. Two families of sets $\mathcal A$ and $\mathcal B$ are called cross-intersecting if $A\cap B \neq \emptyset$ holds for all $A\in\mathcal A$,…

Two families $\mathcal{F}$ and $\mathcal{G}$ are cross-intersecting if every set in $\mathcal{F}$ intersects every set in $\mathcal{G}$. The covering number $\tau(\mathcal{F})$ of a family $\mathcal{F}$ is the minimum size of a set that…

The $2$-layer drawing model is a well-established paradigm to visualize bipartite graphs. Several beyond-planar graph classes have been studied under this model. Surprisingly, however, the fundamental class of $k$-planar graphs has been…

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…

Let $k \geq 3$. We prove the following three bounds for the matching number, $\alpha'(G)$, of a graph, $G$, of order $n$ size $m$ and maximum degree at most $k$. If $k$ is odd, then $\alpha'(G) \ge \left( \frac{k-1}{k(k^2 - 3)} \right) n \,…

Let $n > k > 1$ be integers, $[n] = \{1, \ldots, n\}$. Let $\mathcal F$ be a family of $k$-subsets of~$[n]$. The family $\mathcal F$ is called intersecting if $F \cap F' \neq \emptyset$ for all $F, F' \in \mathcal F$. It is called almost…

For a family of graphs $\mathcal{F}$, the Tur\'{a}n number $ex(n,\mathcal{F})$ is the maximum number of edges in an $n$-vertex graph containing no member of $\mathcal{F}$ as a subgraph. The maximum number of edges in an $n$-vertex connected…

Let $m$, $n$, and $k$ be integers satisfying $0 < k \leq n < 2k \leq m$. A family of sets $\mathcal{F}$ is called an $(m,n,k)$-intersecting family if $\binom{[n]}{k} \subseteq \mathcal{F} \subseteq \binom{[m]}{k}$ and any pair of members of…

Let $\mathcal{F}\subseteq{[n]\choose k}$ be a $t$-intersecting family. Define the $t$-covering number $\tau_t(\mathcal{F})$ of $\mathcal{F}$ as the minimum size of a subset $S$ of $[n]$ with $|S\cap F|\geqslant t$ for each…

This paper resolves two open problems in extremal set theory. For a family $\mathcal{F} \subseteq 2^{[n]}$ and $i, j\in [n]$, we denote $\mathcal{F} (i,\bar{j})=\{F\backslash\{i\}: F\in \mathcal{F}, F\cap\{i,j\}=\{i\}\}$. The sturdiness…

We determine, up to a multiplicative constant, the optimal number of random edges that need to be added to a $k$-graph $H$ with minimum vertex degree $\Omega(n^{k-1})$ to ensure an $F$-factor with high probability, for any $F$ that belongs…

An infinite family of graphs ${\cal F}$ is called feasible if for any pair of integers $(n,m)$, $n \geq 1$, $0 \leq m \leq \binom{n}{2}$, there is a member $G \in {\cal F}$ such that $G$ has $n$ vertices and $m$ edges. We prove that given a…

The well-known Erdos-Ko-Rado Theorem states that if F is a family of k-element subsets of {1,2,...,n} (n>2k-1) such that every pair of elements in F has a nonempty intersection, then |F| is at most $\binom{n-1}{k-1}$. The theorem also…