Block avoiding point sequencings of partial Steiner systems

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