Related papers: Colored Tverberg Theorems for non-prime powers

In this work, we investigate the fewest number of colors needed to guarantee a rainbow solution to the equation $x_1 + x_2 = k x_3$ in $\mathbb{Z}_n$. This value is called the Rainbow number and is denoted by $rb(\mathbb{Z}_n, k)$ for…

We provide an algorithm that verifies the optimal colored Tverberg problem for $10$ points in the plane: Every $10$ points in the plane in color classes of size at most $3$ can be partitioned in $4$ rainbow pieces such that their convex…

Drisko \cite{drisko} proved (essentially) that every family of $2n-1$ matchings of size $n$ in a bipartite graph possesses a partial rainbow matching of size $n$. In \cite{bgs} this was generalized as follows: Any $\lfloor \frac{k+2}{k+1} n…

For an edge-colored graph, a subgraph is called rainbow if all its edges have distinct colors. We show that if $G$ is an edge-colored graph of order $n$ and size $m$ using $c$ colors on its edges, and $m+c\geq \binom{n+1}{2}+k-1$ for a…

Tverberg's theorem bounds the number of points $\mathbb{R}^d$ needed for the existence of a partition into $r$ parts whose convex hulls intersect. If the points are colored with $N$ colors, we seek partitions where each part has at most one…

We study colorful no-dimensional Tverberg-type problems and obtain several optimal results. A colorful no-dimensional Tverberg-type theorem provides a bound on a radius $R$ such that, for any pairwise disjoint $k$-element subsets…

Motivated by a question of Grinblat, we study the minimal number $\mathfrak{v}(n)$ that satisfies the following. If $A_1,\ldots, A_n$ are equivalence relations on a set $X$ such that for every $i\in[n]$ there are at least $\mathfrak{v}(n)$…

Let $n$, $s$, and $k$ be positive integers. We say that a sequence $f_1,\dots,f_s$ of nonnegative integers is satisfying if for any collection of $s$ families $\mathcal F_1,\dots,\mathcal F_s\subseteq [n]^k$ such that $|\mathcal F_i|=f_i$…

Consider the set $\{1,2,\dots,n\} = [n]$ and an equation $eq$. The rainbow number of $[n]$ for $eq$, denoted $\operatorname{rb}([n],eq)$, is the smallest number of colors such that for every exact $\operatorname{rb}([n], eq)$-coloring of…

We prove a quantitative version of the multi-colored Motzkin-Rabin theorem in the spirit of [BDWY12]: Let $V_1,\ldots,V_n \subset R^d$ be $n$ disjoint sets of points (of $n$ `colors'). Suppose that for every $V_i$ and every point $v \in…

For each prime p > 7 we obtain the expression for an upper bound on the minimum number of colors needed to non-trivially color T(2, p), the torus knots of type (2, p), modulo p. This expression is t + 2 l -1 where t and l are extracted from…

For maximal planar graphs of order $n\geq 4$, we prove that a vertex--coloring containing no rainbow faces uses at most $\lfloor\frac{2n-1}{3}\rfloor$ colors, and this is best possible. For maximal graph embedded on the projective plane, we…

For positive integers $t$ and $n$ let $C_t^n$ be the $n$-cube over $t$ elements, that is, the set of ordered $n$-tuples over the alphabet $\{0,\dots, t-1\}$. We address the question of whether a balanced finite coloring of $C_t^n$…

The colorful Helly theorem and Tverberg's theorem are fundamental results in discrete geometry. We prove a theorem which interpolates between the two. In particular, we show the following for any integers $d \geq m \geq 1$ and $k$ a prime…

Many of the strengthenings and extensions of the topological Tverberg theorem can be derived with surprising ease directly from the original theorem: For this we introduce a proof technique that combines a concept of "Tverberg unavoidable…

The topological Tverberg theorem has been generalized in several directions by setting extra restrictions on the Tverberg partitions. Restricted Tverberg partitions, defined by the idea that certain points cannot be in the same part, are…

Suppose that all primes are colored with k colors. Then there exist monochromatic primes p1, p2, p3 such that p1+p2=p3+1.

Let $G = (V, E)$ be an $n$-vertex edge-colored graph. In 2013, H. Li proved that if every vertex $v \in V$ is incident to at least $(n+1)/2$ distinctly colored edges, then $G$ admits a rainbow triangle. We prove that the same hypothesis…

Let $G = (V, E)$ be a graph on $n$ vertices, and let $c: E \to P$, where $P$ is a set of colors. Let $\delta^c(G) = \min_{v \in V} \{ d^{c}(v) \}$ where $d^c(v)$ is the number of colors on edges incident to a vertex $v$ of $G$. In 2011,…

Jungi\'{c} et al (2003) defined $T_{k}$ as the minimal number $t \in \mathbb{N}$ such that there is a rainbow arithmetic progression of length $k$ in every equinumerous $t$-coloring of $[t n]$ for every $n \in \mathbb{N}$. They proved that…