Related papers: Rainbow sets in the intersection of two matroids

Stein proposed the following conjecture: if the edge set of $K_{n,n}$ is partitioned into $n$ sets, each of size $n$, then there is a partial rainbow matching of size $n-1$. He proved that there is a partial rainbow matching of size…

A subgraph of an edge-colored graph is called \emph{rainbow} if all of its edges have distinct colors. There has been much research on the topic of finding a large rainbow matching in a properly edge-colored graph, where a proper…

A perfect matching M in an edge-colored complete bipartite graph K_{n,n} is rainbow if no pair of edges in M have the same color. We obtain asymptotic enumeration results for the number of rainbow matchings in terms of the maximum number of…

By a theorem of Drisko, any $2n-1$ matchings of size $n$ in a bipartite graph have a partial rainbow matching of size $n$. Inspired by discussion of Bar\'at, Gy\'arf\'as and S\'ark\"ozy, we conjecture that if $n$ is odd then the same is…

Given a graph $G$, let $f_{G}(n,m)$ be the minimal number $k$ such that every $k$ independent $n$-sets in $G$ have a rainbow $m$-set. Let $\mathcal{D}(2)$ be the family of all graphs with maximum degree at most two. Aharoni et al. (2019)…

Aharoni and Howard, and, independently, Huang, Loh, and Sudakov proposed the following rainbow version of Erd\H{o}s matching conjecture: For positive integers $n,k,m$ with $n\ge km$, if each of the families $F_1,\ldots, F_m\subseteq…

Consider a matroid $M=(E,\mathcal{I})$ with its elements of the ground set $E$ colored. A rainbow basis is a maximum independent set in which each element receives a different color. The rank of a subset $S$ of $E$, denoted by $r_M(S)$, is…

Rota's Basis Conjecture is a well known problem from matroid theory, that states that for any collection of $n$ bases in a rank $n$ matroid, it is possible to decompose all the elements into $n$ disjoint rainbow bases. Here an asymptotic…

DeVos et al conjectured that if $M$ is a simple, regular matroid and $c$ is a colouring of the elements of $M$ with $r(M)+1$ colours, where each colour class has at least two elements, then $M$ contains a rainbow circuit of size at most…

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…

Aharoni and Howard conjectured that, for positive integers $n,k,t$ with $n\ge k$ and $n\ge t$, if $F_1,\ldots, F_t\subseteq {[n]\choose k}$ such that $|F_i|>{n\choose k}-{n-t+1\choose k}$ for $i\in [t]$ then there exist $e_i\in F_i$ for…

Given a family $\mathcal{I}$ of independent sets in a graph, a rainbow independent set is an independent set $I$ such that there is an injection $\phi\colon I\to \mathcal{I}$ where for each $v\in I$, $v$ is contained in $\phi(v)$. Aharoni,…

Aharoni and Berger conjectured that in any bipartite multigraph that is properly edge-coloured by $n$ colours with at least $n + 1$ edges of each colour there must be a matching that uses each colour exactly once. In this paper we consider…

Let $n$ be a sufficiently large integer with $n\equiv 0\pmod 4$ and let $F_i \subseteq{[n]\choose 4}$ where $i\in [n/4]$. We show that if each vertex of $F_i$ is contained in more than ${n-1\choose 3}-{3n/4\choose 3}$ edges, then $\{F_1,…

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…

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…

A {\it rainbow matching} in an edge-colored graph is a matching in which all the edges have distinct colors. Wang asked if there is a function f(\delta) such that a properly edge-colored graph G with minimum degree \delta and order at least…

Grinblat (2002) asks the following question in the context of algebras of sets: What is the smallest number $\mathfrak v = \mathfrak v(n)$ such that, if $A_1, \ldots, A_n$ are $n$ equivalence relations on a common finite ground set $X$,…

Let $n,m$ be integers such that $1\leq m\leq (n-2)/2$ and let $[n]=\{1,\ldots,n\}$. Let $\mathcal{G}=\{G_1,\ldots,G_{m+1}\}$ be a family of graphs on the same vertex set $[n]$. In this paper, we prove that if for any $i\in [m+1]$, the…

In this paper we consider properly edge-colored graphs, i.e. two edges with the same color cannot share an endpoint, so each color class is a matching. A matching is called \it rainbow \rm if its edges have different colors. The minimum…