Related papers: An Unsure Note on an Un-Schur Problem

What is the minimum number of triangles in a graph of given order and size? Motivated by earlier results of Mantel and Tur\'an, Rademacher solved the first non-trivial case of this problem in 1941. The problem was revived by Erd\H{o}s in…

The anti-Ramsey number, $ar(G, H)$ is the minimum integer $k$ such that in any edge colouring of $G$ with $k$ colours there is a rainbow subgraph isomorphic to $H$, i.e., a copy of $H$ with each of its edges assigned a different colour. The…

Given a coloring of the edges of a multi-hypergraph, a rainbow t-matching is a collection of t disjoint edges, each having a different color. In this note we study the problem of finding a rainbow $t$-matching in an r-partite r-uniform…

It is known that in any $r$-coloring of the edges of a complete $r$-uniform hypergraph, there exists a spanning monochromatic component. Given a Steiner triple system on $n$ vertices, what is the largest monochromatic component one can…

Ramsey's Theorem guarantees for every graph H that any 2-edge-coloring of a sufficiently large complete graph contains a monochromatic copy of H. In 1962, Erdos conjectured that the random 2-edge-coloring minimizes the number of…

The Ramsey number r_k(s,n) is the minimum N such that every red-blue coloring of the k-tuples of an N-element set contains either a red set of size s or a blue set of size n, where a set is called red (blue) if all k-tuples from this set…

For $n\geq s> r\geq 1$ and $k\geq 2$, write $n \rightarrow (s)_{k}^r$ if every hyperedge colouring with $k$ colours of the complete $r$-uniform hypergraph on $n$ vertices has a monochromatic subset of size $s$. Improving upon previous…

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…

The Ramsey number $r(t;\ell)$ is the smallest $n$ such that every $\ell$-coloring of the edges of $K_n$ gives a monochromatic $K_{t}$. In recent years, there have been several improvements on asymptotic lower bounds for these numbers when…

Let G be an n-vertex graph that contains linearly many cherries (i.e., paths on 3 vertices), and let c be a coloring of the edges of the complete graph K_n such that at each vertex every color appears only constantly many times. In 1979,…

The $t$-colored rainbow saturation number $rsat_t(n,F)$ is the minimum size of a $t$-edge-colored graph on $n$ vertices that contains no rainbow copy of $F$, but the addition of any missing edge in any color creates such a rainbow copy.…

The Erd\H{o}s-Rothschild problem from 1974 asks for the maximum number of $s$-edge colourings in an $n$-vertex graph which avoid a monochromatic copy of $K_k$, given positive integers $n,s,k$. In this paper, we systematically study the…

A rainbow subgraph in an edge-coloured graph is a subgraph such that its edges have distinct colours. The minimum colour degree of a graph is the smallest number of distinct colours on the edges incident with a vertex over all vertices.…

The Ramsey number $r(G)$ of a graph $G$ is the smallest integer $n$ such that any $2$ colouring of the edges of a clique on $n$ vertices contains a monochromatic copy of $G$. Determining the Ramsey number of $G$ is a central problem of…

Let $S=\{n_1,n_2,...,n_t\}$ be a finite set of positive integers with $\min(S)\geq 3$ and $t\geq 2$. For any positive integers $s_1,s_2,...,s_t$, we construct a family of 3-uniform bi-hypergraphs ${\cal H}$ with the feasible set $S$ and…

We study an analogue of the Ramsey multiplicity problem for additive structures, in particular establishing the minimum number of monochromatic 3-APs in 3-colorings of $\mathbb{F}_3^n$ as well as obtaining the first non-trivial lower bound…

An edge-colored hypergraph is called \emph{a rainbow hypergraph} if all the colors on its edges are distinct. Given two positive integers $n,r$ and an $r$-uniform hypergraph $\mathcal{G}$, the anti-Ramsey number $ar_r(n,\mathcal{G})$ is…

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.…

A path in an edge-colored graph is called {\em rainbow} if no two edges of it are colored the same. For an $\ell$-connected graph $G$ and an integer $k$ with $1\leq k\leq \ell$, the {\em rainbow $k$-connection number} $rc_k(G)$ of $G$ is…

Erdos and Sos proposed a problem of determining the maximum number F(n) of rainbow triangles in 3-edge-colored complete graphs on n vertices. They conjectured that F(n) = F(a)+ F(b)+F(c)+F(d)+abc+abd+acd+bcd, where a+b+c+d = n and a, b, c,…