Block avoiding point sequencings of partial Steiner systems

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 $\ell \leq 0.0908\,n^{1/2}$.