AI 中文总结
本文证明德萨格极大弧导出的Steiner 2-设计满足Erdős-Ko-Rado性质，回答开放问题，并通过计算找到反例，推广了非典型最大相交族。
AI 中文摘要
本文证明了由德萨格极大弧产生的 Steiner 2-设计中块的最大相交族的 Erdős-Ko-Rado 刻画。这回答了 Goryainov 和 Konstantinova 最近提出的一个问题，并意味着在已知的 Steiner 2-设计中，只有有限多个具有既非典型也非与子设计相关的最大相交族。我们还对 2-(120,8,1) 设计进行了计算研究，并找到了 Godsil 和 Meagher 问题的强反例。最后，我们给出了具有紧对偶弧的 2-(66,6,1) 设计的参数化推广，作为非典型最大相交族。
英文摘要
In this paper, we prove an Erdős-Ko-Rado characterisation of maximum intersecting families of blocks in Steiner $2$-designs arising from Desarguesian maximal arcs. This answers a recent question of Goryainov and Konstantinova, and implies that, among the known Steiner $2$-designs, only finitely many admit a maximum intersecting family that is neither canonical nor associated with a subdesign. We also perform a computational study of $2$-$(120,8,1)$ designs and find strong counterexamples to a problem of Godsil and Meagher. Finally, we give a parametric generalisation of $2$-$(66,6,1)$ designs with tight dual arcs as non-canonical maximum intersecting families.
