Related papers: Monochromatic and Zero-Sum Sets of Nondecreasing D…

We prove that, for every function $f:\mathbb{N} \rightarrow \mathbb{N}$, there is a graph $G$ with uncountable chromatic number such that, for every $k \in \mathbb{N}$ with $k \geq 3$, every subgraph of $G$ with fewer than $f(k)$ vertices…

Alon, Frankl, and Lov\'asz proved a conjecture of Erd\H{o}s that one needs at least $\lceil \frac{n-r(k-1)}{r-1} \rceil$ colors to color the $k$-subsets of $\{1, \dots, n\}$ such that any $r$ of the $k$-subsets that have the same color are…

For positive integers $s$ and $k_1, k_2, ..., k_s$, let $w(k_1,k_2,...,k_s)$ be the minimum integer $n$ such that any $s$-coloring $\{1,2,...,n\} \to \{1,2,...,s\}$ admits a $k_i$-term arithmetic progression of color $i$ for some $i$, $1…

It is known that, for any finite coloring of the naturals, there exists distinct naturals $e_1,e_2,e_3,e_4$ that are the same color such that $e_1+e_2=e_3+e_4$. Consider the following statement which we denote S: For every…

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 proper edge-coloring of the complete graph $K_{n}$ with $C$ colors containing no $k$ vertex-disjoint color…

Given a linear equation $\mathcal{E}$, the $k$-color Rado number $R_k(\mathcal{E})$ is the smallest integer $n$ such that every $k$-coloring of $\{1,2,3,\dots,n\}$ contains a monochromatic solution to $\mathcal E$. The degree of regularity…

Given a graph $G$, its Ramsey number $r(G)$ is the minimum $N$ so that every two-coloring of $E(K_N)$ contains a monochromatic copy of $G$. It was conjectured by Conlon, Fox, and Sudakov that if one deletes a single vertex from $G$, the…

Given any $r$-edge coloring of $K_{n,n}$, how large is the maximum (over all $r$ colors) sized monochromatic subgraph guaranteed to be? We give answers to this problem for $r \leq 8$, when $r$ is a perfect square, and when $r$ is one less…

The typical problem in (generalized) Ramsey theory is to find the order of the largest monochromatic member of a family F (for example matchings, paths, cycles, connected subgraphs) that must be present in any edge coloring of a complete…

Hindman conjectured that any finite partition of $\mathbb{N}$ has a monochromatic $\{x,y,x+y,xy\}$. Recently, Bowen proved the result for all 2-partition. In this paper, we extend Bowen's result to any semiring $(S,+,\cdot)$ such that $Ss$…

In an $r$-coloring of edges of the complete graph on $n$ vertices, how many edges are there in the largest monochromatic connected component? A construction of Gy\'arf\'as shows that for infinitely many values of $r$, there exist colorings…

A recent question in generalized Ramsey theory is that for fixed positive integers $s\leq t$, at least how many vertices can be covered by the vertices of no more than $s$ monochromatic members of the family $\cal F$ in every edge coloring…

It is well-known that in every $r$-coloring of the edges of the complete bipartite graph $K_{n,n}$ there is a monochromatic connected component with at least ${2n\over r}$ vertices. It would be interesting to know whether we can…

We show that any $2$-coloring of $\mathbb{N}$ contains infinitely many monochromatic sets of the form $\{x,y,xy,x+y\},$ and more generally monochromatic sets of the form $\{x_i,\prod x_i,\sum x_i: i\leq k\}$ for any $k\in\mathbb{N}.$ Along…

We show the existence of several infinite monochromatic patterns in the integers obtained as values of suitable symmetric polynomials. The simplest example is the following. For every finite coloring of the natural numbers…

Let $G$ be a finite abelian group written additively, and let $r$ be a multiple of its exponent. The modified Erd\H{o}s-Ginzburg-Ziv constant $\mathsf{s}_r'(G)$ is the smallest integer $s$ such that every zero-sum sequence of length $s$…

For positive integers $c, s \geq 1$, let $M_3(c, s)$ be the least integer such that any set of at least $M_3(c, s)$ points in the plane, no three on a line and colored with $c$ colors, contains a monochromatic triangle with at most $s$…

We consider coloring problems inspired by the theory of anti-Ramsey / rainbow colorings that we generalize to a far extent. Let $\mathcal{F}$ be a hereditary family of graphs; i.e., if $H\in \mathcal{F}$ and $H'\subset H$ then also…

A result of Gy\'arf\'as exactly determines the size of a largest monochromatic component in an arbitrary $r$-coloring of the complete $k$-uniform hypergraph $K_n^k$ when $k\geq 2$ and $r-1\leq k\leq r$. We prove a result which says that if…

We show that there is a rational vector space $V$ such that, whenever $V$ is finitely coloured, there is an infinite set $X$ whose sumset $X+X$ is monochromatic. Our example is the rational vector space of dimension…