Related papers: Polynomial to exponential transition in Ramsey the…

Let $H=(V,E)$ be an $s$-uniform hypergraph of order $n$ and $k\geq 0$ be an integer. A $k$-independent set $S\subseteq H$ is a set of vertices such that the maximum degree in the hypergraph induced by $S$ is at most $k$. Denoted by…

We show that for all $k\geq 4$, $\varepsilon >0$, and $n$ sufficiently large, every $k$-uniform hypergraph on $n$ vertices in which each set of $k-3$ vertices is contained in at least $(5/8 + \varepsilon) \binom{n}{3}$ edges contains a…

For an $r$-uniform hypergraph $H$, let $f(H)$ be the minimum number of complete $r$-partite $r$-uniform subhypergraphs of $H$ whose edge sets partition the edge set of $H$. For a graph $G$, $f(G)$ is the bipartition number of $G$ which was…

A function $f\colon\mathbb R\to\mathbb R$ is called \emph{$k$-monotone} if it is $(k-2)$-times differentiable and its $(k-2)$nd derivative is convex. A point set $P\subset\mathbb R^2$ is \emph{$k$-monotone interpolable} if it lies on a…

Let $n,k,s$ be three integers and $\beta$ be a sufficiently small positive number such that $k\geq 3$, $0<1/n\ll \beta\ll 1/k$ and $ks+k\leq n\leq (1+\beta)ks+k-2$. A $k$-graph is called non-trivial if it has no isolated vertex. In this…

Let $K\_{[k,t]}$ be the complete graph on $k$ vertices from which a set of edges, induced by a clique of order $t$, has been dropped. In this note we give two explicit upper bounds for $R(K\_{[k\_1,t\_1]},\dots, K\_{[k\_r,t\_r]})$ (the…

Let $P(k,n)$ be the set of products of $k$ factors from the set $\{1,\ldots , n\}.$ In 1955, Erd\H{o}s posed the problem of determining the order of magnitude of $|P (2, n)|$ and proved that $|P (2, n)| = o(n^2 )$ for $n \to\infty$. In…

A triangle $T^{(r)}$ in an $r$-uniform hypergraph is a set of $r+1$ edges such that $r$ of them share a common $(r-1)$-set of vertices and the last edge contains the remaining vertex from each of the first $r$ edges. Our main result is that…

The $k$-independence number of a graph is the maximum size of a set of vertices at pairwise distance greater than $k$. A graph is called $k$-partially walk-regular if the number of closed walks of a given length $l\le k$, rooted at a vertex…

In this note, we study the emergence of Hamiltonian Berge cycles in random $r$-uniform hypergraphs. For $r\geq 3$, we prove an optimal stopping-time result that if edges are sequently added to an initially empty $r$-graph, then as soon as…

We generalize the Harary-Sachs theorem to $k$-uniform hypergraphs: the codegree-$d$ coefficient of the characteristic polynomial of a uniform hypergraph ${\cal H}$ can be expressed as a weighted sum of subgraph counts over certain…

Given a graph $H$ and a function $f(n)$, the Ramsey-Tur\'an number $RT(n,H,f(n))$ is the maximum number of edges in an $n$-vertex $H$-free graph with independence number at most $f(n)$. For $H$ being a small clique, many results about…

The $r$-colour Ramsey number $R_r(k)$ is the minimum $n \in \mathbb{N}$ such that every $r$-colouring of the edges of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $K_k$. We prove, for each fixed $r \geqslant 2$,…

In 1973 Bermond, Germa, Heydemann and Sotteau conjectured that if $n$ divides $\binom{n}{k}$, then the complete $k$-uniform hypergraph on $n$ vertices has a decomposition into Hamilton Berge cycles. Here a Berge cycle consists of an…

In this paper, we first study a new extremal problem recently posed by Conlon and Tyomkyn~(arXiv: 2002.00921). Given a graph $H$ and an integer $k\geqslant 2$, let $f_{k}(n,H)$ be the smallest number of colors $c$ such that there exists a…

The upper tail problem in the Erd\H{o}s--R\'enyi random graph $G\sim\mathcal{G}_{n,p}$ asks to estimate the probability that the number of copies of a graph $H$ in $G$ exceeds its expectation by a factor $1+\delta$. Chatterjee and Dembo…

Given graphs $G, H$ and an integer $q \ge 2$, the generalized Ramsey number, denoted $r(G,H,q)$, is the minimum number of colours needed to edge-colour $G$ such that every copy of $H$ receives at least $q$ colours. In this paper, we prove…

A sparse version of Mantel's Theorem is that, for sufficiently large $p$, with high probability (w.h.p.), every maximum triangle-free subgraph of $G(n,p)$ is bipartite. DeMarco and Kahn proved this for $p>K \sqrt{\log n/n}$ for some…

The famous Brown-Erd\H{o}s-S\'os conjecture from 1973 states, in an equivalent form, that for any fixed $\delta>0$ and integer $k\geq 3$ every sufficiently large linear $3$-uniform hypergraph of size $\delta n^2$ contains some $k$ edges…

A hypergraph is simple if it has no loops and no repeated edges, and a hypergraph is linear if it is simple and each pair of edges intersects in at most one vertex. For $n\geq 3$, let $r= r(n)\geq 3$ be an integer and let $\boldsymbol{k} =…