PreprintarXiv licencearXiv
Proving the optimality of the product construction for the intersection--union problem
Abstract
Let m(n,p,q) be the largest size of a family of subsets of [n] whose members pairwise intersect in at least p vertices and omit at least q common vertices. We prove that a product on disjoint sets attains this maximum whenever p,q≥2 and n≥215(p+q−2)3. Together with the second author's result for p=1 or q=1, this establishes Frankl's Intersection--Union Product Conjecture in a wide range.
Related
- Intersection Product Formulas and Integral Dependence
- Product subcomplexes and intersection complexes of weakly special square complexes
- Artinian reduction applied to the Largest Intersection Problem
- On Mattila's intersection problem in the plane: sharp dimension and positive measure of rotated difference sets
- A Riemannian Extension of a Quadrature Surface Free Boundary Problem: Stability and Optimality
BibTeX
@misc{chang2026proving,
title = {{Proving the optimality of the product construction for the intersection--union problem}},
author = {Fan Chang and Peter Frankl and Jian Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25049},
eprint = {2610.25049},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25049},
}
RIS
TY - UNPB
TI - Proving the optimality of the product construction for the intersection--union problem
AU - Fan Chang
AU - Peter Frankl
AU - Jian Wang
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25049
UR - https://arxiv.org/abs/2610.25049
AB - Let m(n,p,q) be the largest size of a family of subsets of [n] whose members pairwise intersect in at least p vertices and omit at least q common vertices. We prove that a product on disjoint sets attains this maximum whenever p,q≥2 and n≥2^15(p+q−2)^3. Together with the second author's result for p=1 or q=1, this establishes Frankl's Intersection--Union Product Conjecture in a wide range.
ER -