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arXiv · 2606.24882

A Resolution of Erd\H{o}s Problems 593 and 1177: Obligatory Triple Systems and Exact Spectra

Eric Li
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Abstract

We resolve Erd\H{o}s Problems #593 and #1177. Problem #593 asks which finite triple systems occur in every uncountably chromatic triple system; the answer is exactly the class generated from private-vertex expansions of finite bipartite graphs by finite disjoint unions and one-point amalgamations. Equivalently, after isolated vertices are removed, a finite triple system is obligatory precisely when it is linear, every hyperedge-node of its Levi graph has an incident bridge, and every Berge cycle is even. The proof uses an exact bridge-trace theorem for complete-rank one-apex sequence lifts. We also prove that, for every uncountable cardinal kappa, there is a linear triple system of chromatic number exactly kappa, with at most 2^{2^mu} vertices when kappa=mu^+. These two ingredients give a class-valued exact avoidance-spectrum dichotomy for every finite forbidden triple system. As a consequence, Erd\H{o}s Problem #1177 has truth values yes, no, and yes. All results of this paper have been formally verified in Lean-4.

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Eric Li. 2026-06-23. A Resolution of Erd\H{o}s Problems 593 and 1177: Obligatory Triple Systems and Exact Spectra. https://arxiv.org/abs/2606.24882

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