For every positive integer $n$, we construct a Hasse diagram with $n$ vertices and independence number $O(n^{3/4})$. Such graphs have chromatic number $\mathrm{\Omega }(n^{1/4})$, which significantly improves the previously best-known constructions of Hasse diagrams having chromatic number $\mathrm{\Theta }(\log n)$. In addition, if we also require girth of at least $k⩾5$, we construct such Hasse diagrams with independence number at most $n^{1-\frac{1}{2k-4}+o(1)}$. The proofs are based on the existence of point-line arrangements in the plane with many incidences and avoids certain forbidden subconfigurations, which we find of independent interest.

中文翻译：

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具有大色数的哈斯图

对于每个正整数 $n$，我们构造了一个哈斯图 $n$ 顶点和独立数 $哦(n^{3/4})$. 这样的图有色数$\mathrm{\Omega }(n^{1/4})$，这显着改进了先前最著名的具有色数的哈斯图的构造 $\mathrm{\Theta }(\mathrm{日志}n)$. 此外，如果我们还要求周长至少为$克⩾5$，我们最多构造这样的具有独立数的哈斯图 $n^{1-\frac{1}{2克-4}+○(1)}$. 证明是基于平面中点线排列的存在性，并避免了某些我们认为具有独立意义的禁止子配置。