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Claim. For a finite $3$-uniform hypergraph $G$ let $F_G(\kappa)$ be the class of $3$-uniform hypergraphs of chromatic number $\kappa$ not containing $G$. The claim answers the three assertions of Problem 1177 in turn: the first holds (a nonempty $F_G(\aleph_1)$ has a member of size at most $2^{2^{\aleph_0}}$), the second fails (there are $G$ and $H$ with $F_G(\aleph_1)$ and $F_H(\aleph_1)$ nonempty but disjoint), and the third holds (if $F_G(\kappa)$ is nonempty for one uncountable $\kappa$ then $F_G(\lambda)$ is nonempty for every uncountable $\lambda$). Both positive answers come from a dichotomy: a finite triple system $G$ lies in every triple system of uncountable chromatic number exactly when it belongs to an explicitly generated class (the private-vertex expansions of finite bipartite graphs, closed under finite disjoint unions and one-point amalgamations), and otherwise $F_G(\kappa)$ is nonempty for every uncountable $\kappa$ (the manuscript's spectrum dichotomy, Corollary 7.1, which states no size bound). At $\aleph_1$ a case analysis (Corollary 7.2) gives a witness of size at most $2^{2^{\aleph_0}}$: either the manuscript's linear triple system $L_{\aleph_1}$ of chromatic number exactly $\aleph_1$, or a lift of size at most $2^{\aleph_0}$. The bound $2^{2^\mu}$ at $\kappa=\mu^+$ is proved for the linear calibration $L_\kappa$ (Theorem 1.2), not for every witness. The counterexample to the second assertion is the pair of two triples sharing a pair and the loose $7$-cycle, which the manuscript shows to be separately avoidable but not jointly. The outcome is mixed, two of the three assertions holding and one failing, so the claim value recorded here is answered. The classification is the manuscript's answer to Problem 593, recorded there. The manuscript and its labeled theorems are on the card of the preprint; the argument has not been independently checked.

Submission note. Posted to erdosproblems.com as a proof claim by Eric Li (account EricLi) on 17 July 2026, giving "GPT-5.5 Pro" as the AI used:

This paper resolves Erdős 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, Erdős Problem #1177 has truth values yes, no, and yes.

Depends on. Li's classification of the obligatory finite triple systems, the same manuscript's answer to Problem 593, whose dichotomy supplies the two positive answers here.

Standing. The claimant is Eric Li, whose preprint "A Resolution of Erdős Problems 593 and 1177: Obligatory Triple Systems and Exact Spectra" was posted to arXiv on 2026-06-23 (arXiv:2606.24882, revised 2026-07-23) and who posted the claim on the site's proof-claims tab on 2026-07-17 as a full proof, naming GPT-5.5 Pro as the system used; the manuscript's acknowledgment names OpenAI's ChatGPT for ideation, proof exploration, programming and formatting, with the author taking responsibility for the contents. The claimant's repository, pinned above at its commit of 2026-07-23, describes itself as a Lean 4 and Mathlib formalization of the arXiv manuscript, developed with Aristotle (Harmonic) by its README's credit, with the theorems problem_1177_part1_unconditional, problem_1177_part2_unconditional and problem_1177_part3_unconditional and the certificate Erdos593.full_resolution_unconditional, and reports the axioms propext, Classical.choice and Quot.sound; those reports are the claimant's own, and the corpus has not built or audited the development, so no formalized evidence is listed. The site's discussion thread carries, from 2026-06-24, a comment reporting the claim and a reply asking for scrutiny of the author's several long claimed resolutions; neither is a review. The site's label is OPEN (page last edited 23 January 2026), no one has reviewed or refereed the result, and the claim stays claimed.