Li 2026 resolution erdos problems 593 1177 obligatory

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corollary_1_3: Li's claimed exact-spectrum dichotomy: for a finite triple system F the uncountable cardinals that are the chromatic number of some F-free triple system form the empty class when F lies in B and the class of all uncountable cardinals otherwise.

corollary_1_4: Li's claimed answers to the three assertions of Erdős Problem 1177 in its exact-chromatic-number form: a small witness exists, two nonempty classes can be disjoint, and nonemptiness at one uncountable cardinal gives it at all of them.

theorem_1_1: Li's claimed classification of the finite triple systems that occur in every triple system of uncountable chromatic number: exactly the members of the class B, equivalently, isolated vertices removed, the linear systems whose Levi graph has a bridge at every hyperedge-node and whose Berge cycles are all even.

theorem_1_2: Li's claimed exact linear calibration: every uncountable cardinal kappa is the chromatic number of some linear triple system, and when kappa is the successor of mu the system can be taken with at most 2^(2^mu) vertices.

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Eric Li, A Resolution of Erdős Problems 593 and 1177: Obligatory Triple Systems and Exact Spectra. arXiv preprint (2026). arXiv:2606.24882. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2606.24882), every other right reserved.

Theorem 1.1 (p. 1) characterizes the obligatory finite triple systems, those forced in every uncountably chromatic triple system, in two further ways. A finite triple system F is obligatory iff it belongs to the class B, the smallest class containing every finite edgeless system and the private-vertex expansion J+ of every finite bipartite graph J (one new vertex added to each edge) and closed under one-point amalgamations and finite disjoint unions. It is obligatory iff, once its isolated vertices are deleted, it is linear, each hyperedge-node of its Levi graph meets a bridge, and all its Berge cycles have even length. Theorem 1.2 (p. 2) gives an exact calibration: every uncountable cardinal kappa is the chromatic number of some linear triple system, which can be taken with at most 2^(2^mu) vertices when kappa = mu+. Corollary 1.3 then yields the class-valued exact spectrum dichotomy (Spec(F) empty if F in B, all uncountable cardinals otherwise), and Corollary 1.4 reads off the answers yes, no, yes for the three clauses of Problem 1177, with G two triples sharing a pair and H the loose 7-cycle witnessing clause (2). The tools are a bridge-trace theorem for one-apex sequence lifts of complete rank, cycle collapse and the correspondence between Berge cycles and cycles of graph derivatives, and a transfinite reservoir recursion, with imported inputs from Erdős-Hajnal-Rothschild, Reiher's bipartite-expansion theorem, the Erdős-Galvin-Hajnal property P and Hajnal-Komjáth (for the loose 7-cycle). For Problem 593 the paper claims the requested classification of forced configurations; for Problem 1177 it claims answers to all three clauses in their exact-chromatic-number form. This is a v1 preprint and remains unverified.

Source: https://arxiv.org/abs/2606.24882.

Read status. Claims checked: Theorems 1.1 and 1.2 and Corollaries 1.3 and 1.4 (pp. 1-2), with their restatements and proofs as Theorem 5.7 (p. 13), Theorem 6.14 and Corollary 6.16 (p. 19) and Corollaries 7.1 and 7.2 (pp. 20-21), were read clause by clause on the printed pages of arXiv v1. The proofs were read but not checked, and the imported theorems were not read in their sources.

Bears on. #593: Theorem 1.1 claims the characterization the problem asks for, of the finite 3-uniform hypergraphs that appear in every 3-uniform hypergraph of uncountable chromatic number; the claim page Li's classification records it as a claim on the problem. #1177: Corollary 1.4 claims the problem's three assertions, in their exact-chromatic-number form, are true, false and true, the second refuted by two triples sharing a pair and the loose 7-cycle; the claim page Li's exact spectra records it as a claim on the problem.

Results.

• Theorem 1.1 (p. 1): a finite triple system is obligatory (occurs in every triple system of uncountable chromatic number) iff it lies in the class B, iff after deleting isolated vertices it is linear, every hyperedge-node of its Levi graph is incident with a bridge, and every Berge cycle has even length.
• Theorem 1.2 (p. 2): for every uncountable kappa there is a linear triple system L_kappa with chi(L_kappa) = kappa; for kappa = mu^+ it may be chosen with |V(L_kappa)| <= 2^(2^mu).
• Corollary 1.3 (p. 2): Spec(F) is empty when F is in B and is the class of all cardinals above aleph_0 when F is not in B.
• Corollary 1.4 (p. 2): the three assertions in the current formulation of Problem 1177 have truth values yes, no, yes; clause (2) is witnessed by two triples sharing a pair and the loose 7-cycle.

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