Related papers: Independent sets in hypergraphs with a forbidden l…

For a $k$-uniform hypergraph $G$ with vertex set $\{1,\ldots,n\}$, the ordered Ramsey number $\operatorname{OR}_t(G)$ is the least integer $N$ such that every $t$-coloring of the edges of the complete $k$-uniform graph on vertex set…

We examine the Maximum Independent Set Problem in an undirected graph. The main result is that this problem can be considered as the solving the same problem in a subclass of the weighted normal twin-orthogonal graphs. The problem is…

A well-known theorem of Erd\H{o}s and Gallai asserts that a graph with no path of length $k$ contains at most $\frac{1}{2}(k-1)n$ edges. Recently Gy\H{o}ri, Katona and Lemons gave an extension of this result to hypergraphs by determining…

We introduce and study a variant of Ramsey numbers for edge-ordered graphs, that is, graphs with linearly ordered sets of edges. The edge-ordered Ramsey number $\overline{R}_e(\mathfrak{G})$ of an edge-ordered graph $\mathfrak{G}$ is the…

Given a vertex-ordered graph $G$, the ordered Ramsey number $r_<(G)$ is the minimum integer $N$ such that every $2$-coloring of the edges of the complete ordered graph $K_N$ contains a monochromatic ordered copy of $G$. Motivated by a…

We provide lower bounds on the connectivity of the independence complexes of hypergraphs. Additionally, we compute the homotopy types of the independence complexes of $d$-uniform properly-connected triangulated hypergraphs.

Let $H=(V,E)$ be an $r$-uniform hypergraph. For each $1 \leq s \leq r-1$, an $s$-path ${\mathcal P}^{r,s}_n$ of length $n$ in $H$ is a sequence of distinct vertices $v_1,v_2,\ldots,v_{s+n(r-s)}$ such that $\{v_{1+i(r-s)},\ldots,…

For fixed integers $r\ge 3,e\ge 3,v\ge r+1$, an $r$-uniform hypergraph is called $\mathscr{G}_r(v,e)$-free if the union of any $e$ distinct edges contains at least $v+1$ vertices. Brown, Erd\H{o}s and S\'{o}s showed that the maximum number…

In a recent work on the bipartite Erd\H{o}s-R\'{e}nyi graph, Do et al. (2023) established upper bounds on the number of connected labeled bipartite graphs with a fixed surplus. We use some recent encodings of bipartite random graphs in…

Ramsey theory is a central and active branch of combinatorics. Although Ramsey numbers for graphs have been extensively investigated since Ramsey's work in the 1930s, there is still an exponential gap between the best known lower and upper…

Let $\mathcal{F}$ be an $r$-uniform hypergraph and $G$ be a multigraph. The hypergraph $\mathcal{F}$ is a Berge-$G$ if there is a bijection $f: E(G) \rightarrow E( \mathcal{F} )$ such that $e \subseteq f(e)$ for each $e \in E(G)$. Given a…

The canonical Ramsey theorem of Erd\H{o}s and Rado implies that for any graph $H$, any edge-coloring (with an arbitrary number of colors) of a sufficiently large complete graph $K_N$ contains a monochromatic, lexicographic, or rainbow copy…

For any positive edge density $p$, a random graph in the Erd\H{o}s-Renyi $G_{n,p}$ model is connected with non-zero probability, since all edges are mutually independent. We consider random graph models in which edges that do not share…

Many problems in extremal graph theory correspond to questions involving homomorphisms into a fixed image graph. Recently, there has been interest in maximizing the number of homomorphisms from graphs with a fixed number of vertices and…

In this paper, we consider the hypothesis testing of correlation between two $m$-uniform hypergraphs on $n$ unlabelled nodes. Under the null hypothesis, the hypergraphs are independent, while under the alternative hypothesis, the hyperdges…

One of the major problems in combinatorics is to determine the number of $r$-uniform hypergraphs ($r$-graphs) on $n$ vertices which are free of certain forbidden structures. This problem dates back to the work of Erd\H{o}s, Kleitman and…

Let $G = (V(G), E(G))$ be a graph. The maximum cardinality of a set $M_k \subseteq E(G)$ such that $M_k$ contains exactly $k$-pairs of adjacent edges of $G$ is called the $k$-nearly edge independence number of $G$, and is denoted by…

We equip the edges of a deterministic graph $H$ with independent but not necessarily identically distributed weights and study a generalized version of matchings (i.e. a set of vertex disjoint edges) in $H$ satisfying the property that…

For an integer $k \geq 2$, an ordered $k$-uniform hypergraph $\mathcal{H}=(H,<)$ is a $k$-uniform hypergraph $H$ together with a fixed linear ordering $<$ of its vertex set. The ordered Ramsey number $\overline{R}(\mathcal{H},\mathcal{G})$…

Let $G$ be a bipartite graph without loops and multiple edges on $v\ge 4$ vertices, which can be drawn on the plane such that any edge intersects at most one other edge. We prove that such graph has at most $3v-8$ edges for even $v\ne 6$…