Erdős Problem #81 — Chordal Clique Partitions: Papers I–III

Abstract

Description This record contains three official author preprints from a research program on Erdős Problem #81, concerning the asymptotic edge clique-partition number of chordal graphs. Each paper is accompanied in the linked GitHub repository by its Lean 4 formalization, frozen sources, reproducibility material, integrity manifests, and documented internal and independent adversarial AI audits. All three current preprint packages closed their recorded external audit process with PASS. This is an intermediate assurance tier, not human peer review or a specialist priority determination. Paper I — _Affine Profile Reduction for Fractional Triangle Packings in Split Graphs_ (v1.3). For every split graph G on n vertices, it proves |E(G)| − 2ν₃*(G) ≤ n²/6 + n, where ν₃*(G) is the fractional triangle-packing number. This is a finite fractional result and does not itself give an integral clique-partition theorem. Paper II — _Complete-Split Extremizers for a Fractional Triangle-Cover Functional on Chordal Graphs_ (v1.2). For every integer n ≥ 1, it determines the exact maximum of |E(G)| − 2τ₃*(G) over n-vertex chordal graphs as ⌊(2n+1)²/24⌋, attained by a complete-split graph. This is an exact fractional-cover extremal theorem and does not itself give an integral clique-partition theorem. Paper III — _Linear-Error Clique Partitions of Split Graphs via Structured Triangle Packing_ (v1.5). It proves cp(G) ≤ n²/6 + O(n) for every split graph G, with sharp quadratic coefficient 1/6. Thus it resolves the split-graph case of Erdős Problem #81 at the conjectured quadratic scale. It does not determine the least uniform linear coefficient and does not resolve the problem for all chordal graphs. Scope. The full chordal-graph case of Erdős Problem #81 remains open. The manuscripts and machine-checked artifacts have not undergone human peer review. Novelty and prior-art conclusions are limited to the documented search corpus. The six deposited PDFs are the English and Spanish editions of Papers I–III. The complete public packages and verification evidence are preserved at the immutable GitHub commit linked below.