Related papers: On sequences covering all rainbow $k$-progressions

Motivated by a problem in theoretical computer science suggested by Wigderson, Alon and Ben-Eliezer studied the following extremal problem systematically one decade ago. Given a graph $H$, let $C(n,H)$ be the minimum number $k$ such that…

We study the multicolor Ramsey numbers for paths and even cycles, $R_k(P_n)$ and $R_k(C_n)$, which are the smallest integers $N$ such that every coloring of the complete graph $K_N$ has a monochromatic copy of $P_n$ or $C_n$ respectively.…

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…

Let $f(n,r,k)$ be the minimal number such that every hypergraph larger than $f(n,r,k)$ contained in $\binom{[n]}{r}$ contains a matching of size $k$, and let $g(n,r,k)$ be the minimal number such that every hypergraph larger than $g(n,r,k)$…

Let $k$ be a positive integer, and $G$ be a $k$-connected graph. An edge-coloured path is \emph{rainbow} if all of its edges have distinct colours. The \emph{rainbow $k$-connection number} of $G$, denoted by $rc_k(G)$, is the minimum number…

Given a graph $H$, let $g(n,H)$ denote the smallest $k$ for which the following holds. We can assign a $k$-colouring $f_v$ of the edge set of $K_n$ to each vertex $v$ in $K_n$ with the property that for any copy $T$ of $H$ in $K_n$, there…

Let $f_r(k)$ be the smallest positive integer $n$ such that every $r$-coloring of $\{1,2,...,n\}$ has a monochromatic solution to the nonlinear equation \[1/x_1+\cdots+1/x_k=1/y,\] where $x_1,...,x_k$ are not necessarily distinct. Brown and…

Let $\mathbb{N}$ be the set of all nonnegative integers. For any positive integer $k$ and any subset $A$ of nonnegative integers, let $r_{1,k}(A,n)$ be the number of solutions $(a_1,a_2)$ to the equation $n=a_1+ka_2$. In 2016, Qu proved…

An edge-colored graph $G$, where adjacent edges may have the same color, is {\it rainbow connected} if every two vertices of $G$ are connected by a path whose edge has distinct colors. A graph $G$ is {\it $k$-rainbow connected} if one can…

For fixed integers $p$ and $q$, let $f(n,p,q)$ denote the minimum number of colors needed to color all of the edges of the complete graph $K_n$ such that no clique of $p$ vertices spans fewer than $q$ distinct colors. Any edge-coloring with…

An edge-coloured graph $G$ is rainbow connected if there exists a rainbow path between any two vertices. A graph $G$ is said to be $k$-rainbow connected if there exists an edge-colouring of $G$ with at most $k$ colours that is rainbow…

The Ramsey number $r_k(s,n)$ is the minimum $N$ such that for every red-blue coloring of the $k$-tuples of $\{1,\ldots, N\}$, there are $s$ integers such that every $k$-tuple among them is red, or $n$ integers such that every $k$-tuple…

We determine a lower gap property for the growth of an unbounded \(\mathbb{Z}\)-valued \(k\)-regular sequence. In particular, if \(f:\mathbb{N}\to\mathbb{Z}\) is an unbounded \(k\)-regular sequence, we show that there is a constant \(c>0\)…

For a set of positive integers $D$, a $k$-term $D$-diffsequence is a sequence of positive integers $a_1<a_2<\cdots<a_k$ such that $a_i-a_{i-1}\in D$ for $i=2,3,\cdots,k$. For $k\in\mathbb{Z}^+$ and $D\subset \mathbb{Z}^+$, we define…

The Ramsey number $r_k(s,n)$ is the smallest integer $N$ such that every $N$-vertex $k$-graph contains either a copy of $K_s^{(k)}$ or an independent set of size $n$. We prove that $r_4(5,n)\ge 2^{2^{cn^{1/7}}}$, where $c>0$ is an absolute…

Given positive integers $p,q$. For any integer $k\ge2$, an edge coloring of the complete $k$-graph $K_n^{(k)}$ is said to be a $(p,q)$-coloring if every copy of $K_p^{(k)}$ receives at least $q$ colors. The Erd\H{o}s-Gy\'{a}rf\'{a}s…

Let P be a finite set of at least two prime numbers, and A the set of positive integers that are products of powers of primes from P. Let F(k) denote the smallest positive integer which cannot be presented as sum of less than k terms of A.…

This paper studies the quantity $p(n,r)$, that is the minimal number of edges of an $n$-uniform hypergraph without panchromatic coloring (it means that every edge meets every color) in $r$ colors. If $r \leq c \frac{n}{\ln n}$ then all…

Addressing a question of Cameron and Erd\Ho s, we show that, for infinitely many values of $n$, the number of subsets of $\{1,2,\ldots, n\}$ that do not contain a $k$-term arithmetic progression is at most $2^{O(r_k(n))}$, where $r_k(n)$ is…

Folkman's Theorem asserts that for each $k \in \mathbb{N}$, there exists a natural number $n = F(k)$ such that whenever the elements of $[n]$ are two-coloured, there exists a set $A \subset [n]$ of size $k$ with the property that all the…