Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Source. The copy read for this card is the complete 28-page published article, printed pp. 259–286. All pages, including the references and added-in-proof note, were visually read. Native text was used for navigation; mathematical formulas were checked against page images. The source record identifies the author-hosted copy read. No second manuscript version or page-by-page equivalence with a different edition is claimed.
Complete proof coverage. The deletion chain comprises the ambient slice identities, elementary entropy estimates, Theorems 2.1–2.2, Proposition 2.3, Theorems 1.4 and 1.1, the full weighted expansion of Theorem 1.5, Theorem 3.1, and Corollary 1.6. The counting chain comprises Lemma 4.1, Theorem 6.1, the sufficient two-tolerance Proposition 7.2, the full complementary-size and containing-set reductions of Propositions 7.1 and 7.3 inside Theorem 1.14, and Theorems 1.15–1.16. The fixed-size concentration lemma supplies the uniform endpoint in the containing-set step.
The separate Section 4 mechanism is retained in Theorem 1.7 rather than replaced by an application of the later counting theorem. The weak delta-system induction, the asymptotic cube-orthogonality argument, and the full spherical averaging deduction are written out in Theorems 1.9, 1.11, and 1.13. Section 10's constant-intersection, parity, modular, affine, and constant-distance proofs are complete, as is the odd-parameter Galvin lower bound and its cyclic-window upper construction. The introduction's large-set lower construction has its own proof.
Exact qualifications. Theorems 1.7 and 1.9 are proved with the necessary eventual thresholds. Theorem 1.13 treats its remaining finite dimensions directly and requires .
The exact numerical Corollaries 1.2–1.3 and 2.4 remain at the explicit scope in numerical_corollaries_scope. Theorem 1.18 is an announcement with its proof explicitly deferred by the source; the later complete simplex chain is located at frankl_1990_partition_property_simplices_euclidean_space.
External and historical scope. Harper's isoperimetric theorem is the exact external input to Theorem 2.1. Invariant rotation measure is the external analytic input to the sphere averaging argument. The source's background statements of Katona, Erdős–Ko–Rado, Frankl–Wilson, Frankl–Füredi, the alternative cited Theorem 3.2, and earlier coding, packing, and prime-power quantitative bounds are not newly proved here. Definitions and historical questions are not counted as proof pages. Conjecture 11.2's added-in-proof resolution is retained as historical source information, without a new status search or a reconstruction of that separately cited proof. No source numerical constant is promoted to a present record.