Block avoiding point sequencings of partial Steiner systems

A partial(n, k, t) _λ-system is a pair (X, B) where X is an n-set of vertices and B is a collection of k-subsets of X called blocks such that each t-set of vertices is a subset of at most λ blocks. A sequencing of such a system is a labelling of its vertices with distinct elements of { 0 , … , n- 1 }. A sequencing is ℓ-block avoiding or, more briefly, ℓ-good if no block is contained in a set of ℓ vertices with consecutive labels. Here we give a short proof that, for fixed k, t and λ, any partial (n, k, t) _λ-system has an ℓ-good sequencing for some ℓ= Θ (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 Verstraëte show that the value of ℓ cannot be increased beyond Θ ((nlog n) ^1/t). A special case of our result shows that every partial Steiner triple system (partial (n, 3 , 2) ₁-system) has an ℓ-good sequencing for each positive integer ℓ⩽0.0908n1/2.