arXiv subjects

Marcelo Tadeu Sales

Publications and source records attributed to Marcelo Tadeu Sales.

3 recordsLinked to original sources

arXiv2021

Independent sets in subgraphs of a shift graph

Erdős, Hajnal and Szemerédi proved that any subset $G$ of vertices of a shift graph $\text{Sh}_{n}^{k}$ has the property that the independence number of the subgraph induced by $G$ satisfies $α(\text{Sh}_{n}^{k}[G])\geq \left(\frac{1}{2}-\varepsilon\right)|G|$, where $\varepsilon\to 0$ as $k\to \infty$. In this note we prove that for $k=2$ and $n \to \infty$ there are graphs $G\subseteq \binom{[n]}{2}$ with $α(\text{Sh}_{n}^{2}[G])\leq \left(\frac{1}{4}+o(1)\right)|G|$, and $\frac{1}{4}$ is best possible. We also consider a related problem for infinite shift graphs.

math.CO↗

arXiv2021

Every Steiner triple system contains an almost spanning d-ary hypertree

In this paper we make a partial progress on the following conjecture: for every $μ>0$ and large enough $n$, every Steiner triple system $S$ on at least $(1+μ)n$ vertices contains every hypertree $T$ on $n$ vertices. We prove that the conjecture holds if $T$ is a perfect $d$-ary hypertree.

math.CO↗

arXiv2021

Colourful matchings

Suppose a committee consisting of three members has to match $n$ candidates to $n$ different positions. Each member of the committee proposes a matching, however the proposed matchings totally disagree, i.e., every candidate is matched to three different positions according to three committee members. All three committee members are very competitive and want to push through as many of their suggestions as possible. Can a committee always find a compromise -- a matching of candidates to positions such that for every committee member a third of all candidates are assigned according to that committee member suggestion? We will consider an asymptotic version of this question and several other variants of similar problem. As an application we will consider an embedding problem -- in particular which configurations large Steiner systems always need to contain.

math.CO↗