AI 中文总结
本文给出Chvátal猜想的一个简短直接的谱证明，并证明其关于投影填充数的加强版本，同时提出两个由数值实验激发的谱Chvátal猜想。
AI 中文摘要
Chvátal猜想：每个下闭集都有一个最大尺寸的交族，该交族是一个星，即包含该族中所有含有某个固定元素的成员。最近，Chang、Liu和Liu给出了这个猜想的一个证明，该证明是更一般结果（如Kleitman猜想和Kahn猜想的一个版本，他们也证明了这些猜想）的推论。我们给出了Chvátal猜想的一个简短、直接的（谱）证明，并证明了关于投影填充数的一个加强版本。我们还提出了两个谱Chvátal猜想，这些猜想是由大量数值实验所激发的。
英文摘要
Chvátal conjectured that every downset has a maximum-size intersecting family which is a star, that is, consists of all members of the downset containing a fixed element. Recently, Chang, Liu and Liu gave a proof of this conjecture, which follows as a corollary of more general results such as Kleitman's conjecture and a version of Kahn's conjecture (which they also prove). We give a short, direct (spectral) proof of Chvátal's conjecture, and a proof of Kleitman's conjecture along the same lines. We also prove a strengthening of Chvátal's conjecture concerning the projection packing number. Finally, we propose two spectral Chvátal conjectures which are motivated by extensive numerical experiments.
Comments9 pages; we have added the short spectral proof of Kleitman's conjecture, along the same lines as the proof of Chvatal's conjecture in the first version of this manuscript
