Related papers: A note on long rainbow arithmetic progressions

Let $\mathcal{G}_n^k=\{G_1,G_2,\ldots,G_k\}$ be a multiset of graphs on vertex set $[n]$ and let $F$ be a fixed graph with edge set $F=\{e_1, e_2,\ldots, e_m\}$ and $k\ge m$. We say ${\mathcal{G}_n^k}$ is rainbow $F$-free if there is no…

Let H(k) be the smallest N such that every finite coloring of [N] contains a monochromatic or rainbow k-term arithmetic progression. Erd\H{o}s and Graham asked whether $H(k)^{1/k}/k \to \infty$ (Problem #190 of the Erd\H{o}s Problems…

A Gallai coloring of a complete graph $K_n$ is an edge coloring without triangles colored with three different colors. A sequence $e_1\ge \dots \ge e_k$ of positive integers is an $(n,k)$-sequence if $\sum_{i=1}^k e_i=\binom{n}{2}$. An…

For a graph $G$, the $k$-colour Ramsey number $R_k(G)$ is the least integer $N$ such that every $k$-colouring of the edges of the complete graph $K_N$ contains a monochromatic copy of $G$. Let $C_n$ denote the cycle on $n$ vertices. We show…

A proper 2-tone $k$-coloring of a graph is a labeling of the vertices with elements from $\binom{[k]}{2}$ such that adjacent vertices receive disjoint labels and vertices distance 2 apart receive distinct labels. The 2-tone chromatic number…

Suppose we are given matchings $M_1,....,M_N$ of size $t$ in some $r$-uniform hypergraph, and let us think of each matching having a different color. How large does $N$ need to be (in terms of $t$ and $r$) such that we can always find a…

A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. The problem of finding rainbow subgraphs goes back to the work of Euler on transversals in Latin squares and was extensively studied since then.…

An edge-coloured cycle is rainbow if the edges have distinct colours. Let $G$ be a graph such that any $k$ vertices lie in a cycle of $G$. The $k$-rainbow cycle index of $G$, denoted by $crx_k(G)$, is the minimum number of colours required…

For a given graph $H$ and $n\geq 1$, let $f(n,H)$ denote the maximum number $c$ for which there is a way to color the edges of the complete graph $K_n$ with $c$ colors such that every subgraph $H$ of $K_n$ has at least two edges of the same…

Strengthening Hadwiger's conjecture, Gerards and Seymour conjectured in 1995 that every graph with no odd $K_t$-minor is properly $(t-1)$-colorable, this is known as the Odd Hadwiger's conjecture. We prove a relaxation of the above…

Let $F$ be a fixed graph. The rainbow Tur\'an number of $F$ is defined as the maximum number of edges in a graph on $n$ vertices that has a proper edge-coloring with no rainbow copy of $F$ (where a rainbow copy of $F$ means a copy of $F$…

The threshold-$k$ metric dimension ($\mathrm{Tmd}_k$) of a graph is the minimum number of sensors -- a subset of the vertex set -- needed to uniquely identify any vertex in the graph, solely based on its distances from the sensors, when the…

Van der Waerden's theorem states that for any positive integers $k$ and $r$, there exists a smallest value $n = w(k,r)$, called the van der Waerden number, such that every $r$-coloring of $\{1,\dots,n\}$ contains a monochromatic $k$-term…

We prove the following asymptotically tight lower bound for $k$-color discrepancy: For any $k \geq 2$, there exists a hypergraph with $n$ hyperedges such that its $k$-color discrepancy is at least $\Omega(\sqrt{n})$. This improves on the…

Consider the set $[m]\times [n] = \{(i,j)\, : 1\le i \le m, 1\le j \le n\}$ and the equation $x_1+x_2 = x_3$, namely $eq$. The \emph{rainbow number of $[m] \times [n]$ for $eq$}, denoted $\text{rb}([m]\times [n],eq)$, is the smallest number…

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.…

We show that there is a red-blue colouring of $[N]$ with no blue 3-term arithmetic progression and no red arithmetic progression of length $e^{C(\log N)^{3/4}(\log \log N)^{1/4}}$. Consequently, the two-colour van der Waerden number…

In 1980, Burr conjectured that every directed graph with chromatic number $2k-2$ contains any oriented tree of order $k$ as a subdigraph. Burr showed that chromatic number $(k-1)^2$ suffices, which was improved in 2013 to $\frac{k^2}{2} -…

For a set of permutations (patterns) $\Pi$ in $S_k$, consider the set of all permutations in $S_n$ that avoid all patterns in $\Pi$. An important problem in current algebraic combinatorics is to find pattern sets $\Pi$ such that the…

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…