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Claim. A finite $3$-uniform hypergraph $F$ occurs in every $3$-uniform hypergraph of chromatic number $>\aleph_0$ exactly when $F$, after its isolated vertices are deleted, is linear (two edges share at most one vertex), every edge-node of its Levi graph (the bipartite incidence graph between the vertices and the edges) is incident with a bridge, and every Berge cycle of $F$ has even length; equivalently, exactly when $F$ lies in the class generated from the private-vertex expansions of finite bipartite graphs (each edge of the graph enlarged by a new vertex of its own) by finite disjoint unions and one-point amalgamations. This is the characterization that Problem 593 asks for. The same paper constructs, for every uncountable cardinal $\kappa$, a linear triple system of chromatic number exactly $\kappa$, on at most $2^{2^\mu}$ vertices when $\kappa=\mu^+$, and combines the two results into a dichotomy: the uncountable chromatic numbers realized by triple systems avoiding a fixed finite $F$ (the paper's spectrum of $F$, defined over the cardinals above $\aleph_0$) form the empty set when $F$ is obligatory and the class of all uncountable cardinals otherwise. From this the paper also reads off answers to Problem 1177, which belong on that problem's page.

The manuscript is Eric Li, "A Resolution of Erdős Problems 593 and 1177: Obligatory Triple Systems and Exact Spectra", arXiv:2606.24882, first version of 2026-06-23 and second version of 2026-07-23 (23 pages), carded at the library card, which records the arXiv record's non-exclusive distribution license.

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, made publicly available on arXiv on 23 June 2026, resolves Erdős Problem #593 (and consequently #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 𝜅, there is a linear triple system of chromatic number exactly 𝜅, with at most 22𝜇 vertices when 𝜅 =𝜇+. These two ingredients give a class-valued exact avoidance-spectrum dichotomy for every finite forbidden triple system. Notes: Note that this paper was made publicly available on arXiv on 23 June 2026, as the first instance of any proof of Erdős Problems #593 and #1177. This proof was also previously discussed in the comments section before the 'Proof Claim' functionality was created, and has now been reflected here. The proof is also being formalised, and the author welcomes others to work together on formalising the results from the proof, regardless of whether his proof is re-derived or repeated by others.

Argument, in outline. The paper's own account of its logical dependence (its Section 1.1) splits the proof of the classification into two halves, neither of which uses the exact-cardinal construction of Theorem 1.2. The negative half, which builds uncountably chromatic hosts avoiding every system that fails the criterion, rests on cycle collapse, the exact bridge-trace theorem for complete-rank one-apex sequence lifts and a cycle-derivative correspondence, with the Erdős–Hajnal–Rothschild nonlinearity obstruction and the Erdős–Hajnal graphs of uncountable chromatic number and large odd girth as imported inputs. The positive half, which forces every system that meets the criterion, rests on expansion pieces, a quotient forest and a running-intersection argument, with Reiher's bipartite-expansion theorem and graph-coloring compactness as imported inputs. The transfinite reservoir recursion belongs to the exact calibration of Theorem 1.2, which the spectrum dichotomy and the answers to Problem 1177 use.

Standing. The claimant is Eric Li, who posted the preprint on arXiv on 2026-06-23 (a comment in the site's discussion thread reported it on 2026-06-24) and filed the result on the site's proof-claims page on 2026-07-17; the claim's entry names the system GPT-5.5 Pro. A Lean 4 development in the repository ericlisg/erdos-593-1177-lean declares itself a formalization of this resolution, developed with the system Aristotle (Harmonic); its README names the theorems Erdos593.full_resolution_unconditional and classification_unconditional, with three further theorems for Problem 1177, and reports a successful build last verified with the axioms propext, Classical.choice and Quot.sound and no sorry; the second arXiv version's comment says that all results are formally verified in Lean 4. This corpus has not built, kernel-checked or audited the development, and no fidelity review of the Lean statement exists, so the page lists no formalized evidence. The site shows no verdict and its label is OPEN; the manuscript is not refereed and no one is recorded as having reviewed it, so the claim stays claimed. The site's three comments on the claim concern only the filing of a claim on Problem 1177. The same characterization is claimed, from a separate manuscript with its own Lean development, by Petkov's alternative proof, which credits this preprint as the first complete proof and reuses its framework for the converse; the two pending claims agree.