Related papers: Multicolor Erd\H{o}s--Rogers Functions

We give asymptotically optimal constructions in generalized Ramsey theory using results about conflict-free hypergraph matchings. For example, we present an edge-coloring of $K_{n,n}$ with $2n/3 + o(n)$ colors such that each $4$-cycle…

We extend two well-known results in Ramsey theory from from $K_n$ to arbitrary $n$-chromatic graphs. The first is a note of Erd\H os and Rado stating that in every 2-coloring of the edges of $K_n$ there is a monochromatic tree on $n$…

Let $K_{n}^{r}$ denote the complete $r$-uniform hypergraph on $n$ vertices. A matching $M$ in a hypergraph is a set of pairwise vertex disjoint edges. Recent Ramsey-type results rely on lemmas about the size of monochromatic matchings. A…

Gerards and Seymour conjectured that every graph with no odd $K_t$ minor is $(t-1)$-colorable. This is a strengthening of the famous Hadwiger's Conjecture. Geelen et al. proved that every graph with no odd $K_t$ minor is $O(t\sqrt{\log…

We consider the Erd{\H{o}}s - Faber - Lov\'{a}sz (EFL) conjecture for hypergraphs. This paper gives an upper bound for the chromatic number of $r$ regular linear hypergraphs $\textbf{H}$ of size $n$. If $r \ge 4$, $\chi (\textbf{H}) \le…

This paper focuses on extensions of the classic Erd\H{o}s-Gallai Theorem for the set of weighted function of each edge in a graph. The weighted function of an edge $e$ of an $n$-vertex uniform hypergraph $\mathcal{H}$ is defined to a…

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…

Let the integers $1,\ldots,n$ be assigned colors. Szemer\'edi's theorem implies that if there is a dense color class then there is an arithmetic progression of length three in that color. We study the conditions on the color classes forcing…

Motivated by recent work on majority edge-colourings of graphs, we initiate the study of the corresponding problem for hypergraphs. First, sharpening the probabilistic argument by a $KL$ large-deviation estimate, we obtain a sufficient…

Let $m(n,r)$ denote the minimal number of edges in an $n$-uniform hypergraph which is not $r$-colorable. For the broad history of the problem see [RaiSh]. It is known that for a fixed $n$ the sequence \[ \frac{m(n,r)}{r^n} \] has a limit.…

Recently, Alon and Frankl (JCTB, 2024) determined the maximum number of edges in $K_{\ell+1}$-free $n$-vertex graphs with bounded matching number. For integers $\ell\ge r \ge 2$, the family $\mathcal{K}_{\ell+1}^{r}$ consists of all…

The Erd\H{o}s-Szekeres Theorem stated in terms of graphs says that any red-blue coloring of the edges of the ordered complete graph $K_{rs+1}$ contains a red copy of the monotone increasing path with $r$ edges or a blue copy of the monotone…

The Ruzsa-Szemer\'{e}di $(6,3)$-problem can be equivalently stated as determining the maximum number of edge-disjoint triangles on $n$ vertices such that no triangle is formed by edges from three distinct triangle-copies. Gowers and Janzer…

A subgraph of an edge-colored graph is called \emph{rainbow} if all of its edges have distinct colors. There has been much research on the topic of finding a large rainbow matching in a properly edge-colored graph, where a proper…

For a positive integer $k$ and a graph $H$, the $k$-color induced size-Ramsey number $\hat{R}_{\mathrm{ind}}(H, k)$ is the minimum integer $m$ for which there exists a graph $G$ with $m$ edges such that for every $k$-edge coloring of $G$,…

For a digraph $G$, let $f(G)$ be the maximum chromatic number of an acyclic subgraph of $G$. For an $n$-vertex digraph $G$ it is proved that $f(G) \ge n^{5/9-o(1)}s^{-14/9}$ where $s$ is the bipartite independence number of $G$, i.e., the…

The multicolor Ramsey number problem asks, for each pair of natural numbers $\ell$ and $t$, for the largest $\ell$-coloring of a complete graph with no monochromatic clique of size $t$. Recent works of Conlon-Ferber and Wigderson have…

We focus on two hypergraph Ramsey problems. First, we consider the Erd\H{o}s-Hajnal function $r_k(k+1,t;n)$. In 1972, Erd\H{o}s and Hajnal conjectured that the tower growth rate of $r_k(k+1,t;n)$ is $t-1$ for each $2\le t\le k$. To finish…

The well-known Ramsey number $r(t,u)$ is the smallest integer $n$ such that every $K_t$-free graph of order $n$ contains an independent set of size $u$. In other words, it contains a subset of $u$ vertices with no $K_2$. Erd{\H o}s and…

We study a variant of the Erd\H{o}s Matching Problem in random hypergraphs. Let $\mathcal{K}_p(n,k)$ denote the Erd\H{o}s-R\'enyi random $k$-uniform hypergraph on $n$ vertices where each possible edge is included with probability $p$. We…