Related papers: Colored Tverberg Theorems for non-prime powers

For a set of positive integers $A \subseteq [n]$, an $r$-coloring of $A$ is rainbow sum-free if it contains no rainbow Schur triple. In this paper we initiate the study of the rainbow Erd\H{o}s-Rothchild problem in the context of sum-free…

Let $G$ be an edge-colored graph. We use $e(G)$ and $c(G)$ to denote the number of edges of $G$ and the number of colors appearing on $E(G)$, respectively. For a vertex $v\in V(G)$, the \emph{color neighborhood} of $v$ is defined as the set…

A \textit{rainbow subgraph} of an edge-colored graph is a subgraph whose edges have distinct colors. The \textit{color degree} of a vertex $v$ is the number of different colors on edges incident to $v$. We show that if $n$ is large enough…

The rainbow connection number of a graph G is the least number of colours in a (not necessarily proper) edge-colouring of G such that every two vertices are joined by a path which contains no colour twice. Improving a result of Caro et al.,…

Let $f(n,p,q)$ be the minimum number of colors necessary to color the edges of $K_n$ so that every $K_p$ is at least $q$-colored. We improve current bounds on the {7/4}n-3$, slightly improving the bound of Axenovich. We make small…

We use the probabilistic method to obtain versions of the colorful Carath\'eodory theorem and Tverberg's theorem with tolerance. In particular, we give bounds for the smallest integer $N=N(t,d,r)$ such that for any $N$ points in $R^d$,…

We show that for any integer $t \ge 2$, every properly edge colored $n$-vertex graph with average degree at least $(\log n)^{2+o(1)}$ contains a rainbow subdivision of a complete graph of size $t$. Note that this bound is within $(\log…

In Euclidean Ramsey Theory usually we are looking for monochromatic configurations in the Euclidean space, whose points are colored with a fixed number of colors. In the canonical version, the number of colors is arbitrary, and we are…

K\"onig's edge coloring theorem says that a bipartite graph with maximal degree $n$ has an edge coloring with no more than $n$ colors. We explore the computability theory and Reverse Mathematics aspects of this theorem. Computable bipartite…

In this paper we present a proof of the BMZ Reduction Lemma with a motivational perspective, and state this lemma for maps to manifolds using the classical definition of cohomological dimension. The lemma, proved and utilized in [4], gives…

A rainbow matching in an edge-colored graph is a matching in which no two edges have the same color. The color degree of a vertex v is the number of different colors on edges incident to v. Kritschgau [Electron. J. Combin. 27(2020)] studied…

A path in an edge-colored graph is said to be rainbow if no color repeats on it. An edge-colored graph is said to be rainbow $k$-connected if every pair of vertices is connected by $k$ internally disjoint rainbow paths. The rainbow…

A simpler proof of the four color theorem is presented. The proof was reached using a series of equivalent theorems. First the maximum number of edges of a planar graph is obatined as well as the minimum number of edges for a complete…

We show that if a multigraph $G$ with maximum edge-multiplicity of at most $\frac{\sqrt{n}}{\log^2 n}$, is edge-coloured by $n$ colours such that each colour class is a disjoint union of cliques with at least $2n + o(n)$ vertices, then it…

Given two graphs $G$ and $H$, the {\it rainbow number} $rb(G,H)$ for $H$ with respect to $G$ is defined as the minimum number $k$ such that any $k$-edge-coloring of $G$ contains a rainbow $H$, i.e., a copy of $H$, all of whose edges have…

Let $g$ be sufficiently large, $b\in\{0,\ldots,g-1\}$, and $\mathcal{S}_b$ be the set of integers with no digit equal to $b$ in their base $g$ expansion. We prove that every sufficiently large odd integer $N$ can be written as $p_1 + p_2 +…

Let $G$ be a simple $n$-vertex graph and $c$ be a colouring of $E(G)$ with $n$ colours, where each colour class has size at least $2$. We prove that $(G,c)$ contains a rainbow cycle of length at most $\lceil \frac{n}{2} \rceil$, which is…

We study an anti-Ramsey extension of the classical Corr\'{a}di--Hajnal Theorem: how many colors are needed to color the complete graph on $n$ vertices in order to guarantee a rainbow copy of $t K_{3}$, that is, $t$ vertex-disjoint…

Suppose that $k$ is a non-negative integer and a bipartite multigraph $G$ is the union of $$N=\left\lfloor \frac{k+2}{k+1}n\right\rfloor -(k+1)$$ matchings $M_1,\dots,M_N$, each of size $n$. We show that $G$ has a rainbow matching of size…

Let [n]=\{1,\,2,...,\,n\} be colored in k colors. A rainbow AP(k) in [n] is a k term arithmetic progression whose elements have diferent colors. Conlon, Jungic and Radoicic [10] had shown that there exists an equinumerous 4-coloring of [4n]…