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A graph $ G $ is said to be $ (H;k) $-vertex stable if $ G $ contains a~subgraph isomorphic to $ H $ even after removing any $ k $ of its vertices alongside with their incident edges. We will denote by $ \text{stab}(H;k) $ the minimum size…

A famous conjecture of Erd\H{o}s asserts that for $k\ge 3$, the maximum number of edges in an $n$-vertex $k$-uniform hypergraph without $s+1$ pairwise disjoint edges is $\max\{\binom{n}{k}-\binom{n-s}{k},\binom{sk+k-1}{k}\}$. This problem…

We show that for an infinitely many natural numbers $k$ there are $k$-uniform hypergraphs which admit a `rescaling phenomenon' as described in [9]. More precisely, let $\mathcal{A}(k,I, n)$ denote the class of $k$-graphs on $n$ vertices in…

The chromatic threshold $\delta_\chi(H)$ of a graph $H$ is the infimum of $d>0$ such that the chromatic number of every $n$-vertex $H$-free graph with minimum degree at least $d n$ is bounded by a constant depending only on $H$ and $d$.…

Let $\text{Tr}(n,m,k)$ denote the largest number of distinct projections onto $k$ coordinates guaranteed in any family of $m$ binary vectors of length $n$. The classical Sauer-Perles-Shelah Lemma implies that $\text{Tr}(n, n^r, k) = 2^k$…

A classical theorem of Ahlswede and Katona determines the maximum density of the $2$-edge star in a graph with a given edge density. Motivated by its application in hypergraph Tur\'{a}n problems, we establish a refinement of their result…

We prove a stability version of Harper's cube vertex isoperimetric inequality, showing that subsets of the cube with vertex boundary close to the minimum possible are close to (generalised) Hamming balls. Furthermore, we obtain a local…

Let $\mathcal{F}$ be a family of $r$-uniform hypergraphs. The feasible region $\Omega(\mathcal{F})$ of $\mathcal{F}$ is the set of points $(x,y)$ in the unit square such that there exists a sequence of $\mathcal{F}$-free $r$-uniform…

Let $H=(V,E)$ be a hypergraph, where $V$ is a set of vertices and $E$ is a set of non-empty subsets of $V$ called edges. If all edges of $H$ have the same cardinality $r$, then $H$ is a $r$-uniform hypergraph; if $E$ consists of all…

In 1999, Katona and Kierstead conjectured that if a $k$-uniform hypergraph $\cal H$ on $n$ vertices has minimum co-degree $\lfloor \frac{n-k+3}{2}\rfloor$, i.e., each set of $k-1$ vertices is contained in at least $\lfloor…

A copy of a hypergraph $F$ is called an $F$-copy. Let $K_k^r$ denote the complete $r$-uniform hypergraph whose vertex set is $[k] = \{1, \dots, k\}$ (that is, the edges of $K_k^r$ are the $r$-element subsets of $[k]$). Given an $r$-uniform…

An $\mathcal{F}$-saturated $r$-graph is a maximal $r$-graph not containing any member of $\mathcal{F}$ as a subgraph. Let $\mathcal{K}_{\ell + 1}^{r}$ be the collection of all $r$-graphs $F$ with at most $\binom{\ell+1}{2}$ edges such that…

A Berge-path of length $k$ in a hypergraph $\mathcal{H}$ is a sequence $v_1,e_1,v_2,e_2,\dots,v_{k},e_k,v_{k+1}$ of distinct vertices and hyperedges with $v_{i},v_{i+1} \in e_i$, for $i \le k$. F\"uredi, Kostochka and Luo, and independently…

Given a graph $F$, the expansion $F^{(r)}$ of $F$ is defined as the $r$-uniform hypergraph obtained from $F$ by adding a set of $(r-2)$ distinct new vertices to each edge of $F$. In this paper, we investigate spectral stability results for…

Ryser's Conjecture states that any $r$-partite $r$-uniform hypergraph has a vertex cover of size at most $r - 1$ times the size of the largest matching. For $r = 2$, the conjecture is simply K\"onig's Theorem and every bipartite graph is a…

For every positive integer $t$ we construct a finite family of triple systems ${\mathcal M}_t$, determine its Tur\'{a}n number, and show that there are $t$ extremal ${\mathcal M}_t$-free configurations that are far from each other in…

For a $k$-uniform hypergraph $F$ we consider the parameter $\Theta(F)$, the minimum size of a clique cover of the of $F$. We derive bounds on $\Theta(F)$ for $F$ belonging to various classes of hypergraphs.

The "infamous upper tail problem" for $r$-uniform hypergraphs is to estimate the probability that the number of copies of a fixed hypergraph $H$ in a large binomial $r$-uniform hypergraph $\boldsymbol{G}$ exceeds its expectation by a…

In this paper, we extend and refine previous Tur\'an-type results on graphs with a given circumference. Let $W_{n,k,c}$ be the graph obtained from a clique $K_{c-k+1}$ by adding $n-(c-k+1)$ isolated vertices each joined to the same $k$…

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…