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 Verstraëte 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 \(\ell \leqslant 0.0908\,n^{1/2}\).

部分 \((n,k,t)_\lambda \) -系统是一对\((X,{\mathcal {B}})\)其中X是一个n -顶点集和\({\ mathcal {B}}\)是称为块的X的k个子集的集合，使得每个t个顶点集最多是\(\lambda \)块的子集。这种系统的排序是用\(\{0,\ldots ,n-1\}\)的不同元素标记其顶点。排序是\(\ell \) -避免块，或者更简单地说，\(\ell \) -如果没有块包含在一组具有连续标签的\(\ell \)顶点中，则很好。这里我们给出一个简短的证明，对于固定的k、t和\(\lambda \)，任何部分\((n,k,t)_\lambda \) -系统都有一个\(\ell \) -良好的排序对于一些\(\ell =\Theta (n^{1/t})\)随着n变大。这改善了布莱克本和埃齐翁以及斯廷森和维奇的结果。我们的结果可能在\(k=t+1\)的情况下最感兴趣，其中 Kostochka、Mubayi 和 Verstraëte 的结果表明\(\ell \)的值不能增加到超过\(\Theta ((n \log n)^{1/t})\)。我们的结果的一个特殊情况表明，每个部分 Steiner 三元系统（部分\((n,3,2)_1\) -system）对于每个正整数\( \ ell\ leqslant 0.0908\,n^{1/2}\)。