Block avoiding point sequencings of partial Steiner systems

Abstract

AbstractA partial$$(n,k,t)_\lambda $$ ( n , k , t ) λ -system is a pair $$(X,{\mathcal {B}})$$ ( X , B ) where X is an n-set of vertices and $${\mathcal {B}}$$ B is a collection of k-subsets of X called blocks such that each t-set of vertices is a subset of at most $$\lambda $$ λ blocks. A sequencing of such a system is a labelling of its vertices with distinct elements of $$\{0,\ldots ,n-1\}$$ { 0 , … , n - 1 } . A sequencing is $$\ell $$ ℓ -block avoiding or, more briefly, $$\ell $$ ℓ -good if no block is contained in a set of $$\ell $$ ℓ vertices with consecutive labels. Here we give a short proof that, for fixed k, t and $$\lambda $$ λ , any partial $$(n,k,t)_\lambda $$ ( n , k , t ) λ -system has an $$\ell $$ ℓ -good sequencing for some $$\ell =\Theta (n^{1/t})$$ 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 $$k=t+1$$ k = t + 1 where results of Kostochka, Mubayi and Verstraëte show that the value of $$\ell $$ ℓ cannot be increased beyond $$\Theta ((n \log n)^{1/t})$$ Θ ( ( n log n ) 1 / t ) . A special case of our result shows that every partial Steiner triple system (partial $$(n,3,2)_1$$ ( n , 3 , 2 ) 1 -system) has an $$\ell $$ ℓ -good sequencing for each positive integer $$\ell \leqslant 0.0908\,n^{1/2}$$ ℓ ⩽ 0.0908 n 1 / 2 .