Block avoiding point sequencings of partial Steiner systems

TL;DR

This paper proves that for large partial Steiner systems, there exists a vertex labeling avoiding long consecutive blocks, with the size of such blocks growing proportionally to the square root of the number of vertices.

Contribution

The authors establish a new bound showing the existence of $ ext{ell}$-good sequencings in partial Steiner systems, improving previous results and providing specific bounds for Steiner triples.

Findings

Existence of $ ext{ell}$-good sequencing for $ ext{ell}= heta(n^{1/t})$ in partial systems.

Improved bounds over previous work by Blackburn, Etzion, Stinson, and Veitch.

Specific bound for partial Steiner triples: $ ext{ell} extless 0.0908 n^{1/2}$.

Abstract

A partial $(n,k,t)_\lambda$-system is a pair $(X,\mathcal{B})$ where $X$ is an $n$-set of vertices and $\mathcal{B}$ is a collection of $k$-subsets of $X$ called blocks such that each $t$-set of vertices is a subset of at most $\lambda$ blocks. A sequencing of such a system is a labelling of its vertices with distinct elements of $\{0,\ldots,n-1\}$. A sequencing is $\ell$-block avoiding or, more briefly, $\ell$-good if no block is contained in a set of $\ell$ vertices with consecutive labels. Here we give a short proof that, for fixed $k$, $t$ and $\lambda$, any partial $(n,k,t)_\lambda$-system has an $\ell$-good sequencing for some $\ell=\Theta(n^{1/t})$ as $n$ becomes large. This improves on results of Blackburn and Etzion, and of Stinson and Veitch. Our result is perhaps of most interest in the case $k=t+1$ where results of Kostochka, Mubayi and Verstra\"{e}te show that the value of $\ell$ cannot be increased beyond $\Theta((n \log n)^{1/t})$. A special case of our result shows that every partial Steiner triple system (partial $(n,3,2)_1$-system) has an $\ell$-good sequencing for each positive integer $\ell \leq 0.0908\,n^{1/2}$.