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

Let $G$ be a graph. For a given positive integer $d$, let $f_G(d)$ denote the largest integer $t$ such that in every coloring of the edges of $G$ with two colors there is a monochromatic subgraph with minimum degree at least $d$ and order…

A $k$-term arithmetic progression ($k$-AP) in a graph $G$ is a list of vertices such that each consecutive pair of vertices is the same distance apart. If $c$ is a coloring function of the vertices of $G$ and a $k$-AP in $G$ has each vertex…

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…

Let $k\in \mathbb{N}$ and $s\geq k(\log k+3.20032)$. Let $\mathbb{N}_{0}^{k}$ be the set of $k$-th powers of nonnegative integers. Assume that $\psi$ is an increasing function tending to infinity with $\psi(x)=o(\log x)$ and satifying some…

The canonical van der Waerden theorem asserts that, for sufficiently large $n$, every colouring of $[n]$ contains either a monochromatic or a rainbow arithmetic progression of length $k$ ($k$-AP, for short). In this paper, we determine the…

For a given class $\mathcal{C}$ of graphs and given integers $m \leq n$, let $f_\mathcal{C}(n,m)$ be the minimal number $k$ such that every $k$ independent $n$-sets in any graph belonging to $\mathcal{C}$ have a (possibly partial) rainbow…

Let k be a positive integer. A sequence s over an n-element alphabet A is called a k-radius sequence if every two symbols from A occur in s at distance of at most k. Let f_k(n) denote the length of a shortest k-radius sequence over A. We…

The van der Waerden number W(k,2) is the smallest integer n such that every 2-coloring of 1 to n has a monochromatic arithmetic progression of length k. The existence of such an n for any k is due to van der Waerden but known upper bounds…

The Van der Waerden number $W(k,r)$ denotes the smallest $n$ such that whenever $[n]$ is $r$--colored there exists a monochromatic arithmetic progression of length $k$. Similarly, the Hilbert cube number $h(k,r)$ denotes the smallest $n$…

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. A construction is given which…

Let $k$ and $n$ be positive integers, $n>k$. Define $r(n,k)$ to be the minimum positive value of $$ |\sqrt{a_1} + ... + \sqrt{a_k} - \sqrt{b_1} - >... -\sqrt{b_k} | $$ where $ a_1, a_2, ..., a_k, b_1, b_2, ..., b_k $ are positive integers…

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…

Given a graph $H$, the maximal anti-Ramsey function $f(n,e,H)$ denotes the minimum integer $f$ for which there exists an $n$-vertex graph $G$ with at least $e$ edges admitting an edge-coloring with $f$ colors in which each copy of $H$ in…

For positive integers $n$ and $k$, the \emph{anti-van der Waerden number} of $\mathbb{Z}_n$, denoted by $aw(\mathbb{Z}_n,k)$, is the minimum number of colors needed to color the elements of the cyclic group of order $n$ and guarantee there…

Let $m(n,r)$ denote the minimal number of edges in an $n$-uniform hypergraph which is not $r$-colorable. It is known that for a fixed $n$ one has \[ c_n r^n < m(n,r) < C_n r^n. \] We prove that for any fixed $n$ the sequence $a_r :=…

The locating-chromatic number of a graph $G$ is the smallest integer $n$, such that $G$ has a proper $n$-coloring $c$ and all vertices have different vectors of distances to the colors generated by $c$. We study the asymptotic value of the…

Let $R_k(H;K_m)$ be the smallest number $N$ such that every coloring of the edges of $K_{N}$ with $k+1$ colors has either a monochromatic $H$ in color $i$ for some $1\leqslant i\leqslant k$, or a monochromatic $K_{m}$ in color $k+1$. In…

Let $G$ be a nontrivial connected graph with an edge-coloring $c:E(G)\rightarrow \{1,2,\ldots,q\},$ $q\in \mathbb{N}$, where adjacent edges may be colored the same. A tree $T$ in $G$ is called a $rainbow~tree$ if no two edges of $T$ receive…

A sequence of positive integers $w_1,w_2,...,w_n$ is called an ascending wave if $w_{i+1}-w_i \geq w_i - w_{i-1}$ for $2 \leq i \leq n-1$. For integers $k,r\geq1$, let $AW(k;r)$ be the least positive integer such that under any $r$-coloring…

Let $\mathcal{A}=(a_n)_{n\in\mathbb{N}_+}$ be a sequence of positive integers. Let $p_\mathcal{A}(n,k)$ denote the number of multi-color partitions of $n$ into parts in $\{a_1,\ldots,a_k\}$. We examine several arithmetic properties of the…