arXiv subjects

David Munhá Correia

Publications and source records attributed to David Munhá Correia.

23 records · Page 2Linked to original sources

arXiv2021

Short proofs of rainbow matching results

A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back to the work of Euler on Latin squares and has been the focus of extensive research ever since. Many conjectures in this area roughly say that 'every edge coloured graph of a certain type contains a rainbow matching using every colour'. In this paper we introduce a versatile 'sampling trick', which allows us to obtain short proofs of old results as well as to solve asymptotically some well known conjectures. - We give a simple proof of Pokrovskiy's asymptotic version of the Aharoni-Berger conjecture with greatly improved error term. - We give the first asymptotic proof of the 'non-bipartite' Aharoni-Berger conjecture, solving two conjectures of Aharoni, Berger, Chudnovsky and Zerbib. - We give a very short asymptotic proof of Grinblat's conjecture (first obtained by Clemens, Ehrenmüller, and Pokrovskiy). Furthermore, we obtain a new asymptotically tight bound for Grinblat's problem as a function of edge multiplicity of the corresponding multigraph. - We give the first asymptotic proof of a 30 year old conjecture of Alspach.

math.CO↗

arXiv2021

Flattening rank and its combinatorial applications

Given a $d$-dimensional tensor $T:A_1\times\dots\times A_d\rightarrow \mathbb{F}$ (where $\mathbb{F}$ is a field), the $i$-flattening rank of $T$ is the rank of the matrix whose rows are indexed by $A_{i}$, columns are indexed by $B_{i}=A_1\times\dots\times A_{i-1}\times A_{i+1}\times\dots\times A_{d}$ and whose entries are given by the corresponding values of $T$. The max-flattening rank of $T$ is defined as $\text{mfrank}(T)=\max_{i\in [d]}\text{frank}_{i}(T)$. A tensor $T:A^{d}\rightarrow\mathbb{F}$ is called semi-diagonal, if $T(a,\dots,a)\neq 0$ for every $a\in A$, and $T(a_{1},\dots,a_{d})=0$ for every $a_{1},\dots,a_{d}\in A$ that are all distinct. In this paper we prove that if $T:A^{d}\rightarrow\mathbb{F}$ is semi-diagonal, then $\text{mfrank}(T)\geq \frac{|A|}{d-1}$, and this bound is the best possible. We give several applications of this result, including a generalization of the celebrated Frankl-Wilson theorem on forbidden intersections. Also, addressing a conjecture of Aharoni and Berger, we show that if the edges of an $r$-uniform multi-hypergraph $\mathcal{H}$ are colored with $z$ colors such that each colorclass is a matching of size $t$, then $\mathcal{H}$ contains a rainbow matching of size $t$ provided $z>(t-1)\binom{rt}{r}$. This improves previous results of Alon and Glebov, Sudakov and Szabó.

math.CO↗

arXiv2021

Powers of paths in tournaments

In this short note we prove that every tournament contains the $k$-th power of a directed path of linear length. This improves upon recent results of Yuster and of Girão. We also give a complete solution for this problem when $k=2$, showing that there is always a square of a directed path of length $\lceil 2n/3 \rceil-1$, which is best possible.

math.CO↗

arXiv2020

A Note on Powers of Paths in Tournaments

In this note we show that every tournament on $n$ vertices contains the $k$-th power of a directed path of length $n/2^{6k+7}$, which improves upon the recent bound of Scott and Korándi of $n/2^{2^{3k}}$. By doing so, we get an inverse exponential dependence on $k$, which is best possible as Yuster recently showed an upper bound of $kn/{2^{k/2}}$.

math.CO↗

arXiv2020

Full rainbow matchings in equivalence relations

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 has a full rainbow matching, that is, a matching where each colour appears exactly once. This asymptotically solves a question raised by Clemens, Ehrenmüller and Pokrovskiy, and is related to problems on algebras of sets studied by Grinblat in [Grinblat 2002].

math.CO↗