Sizes of cross-intersecting families
Abstract
The size of cross-intersecting families cal F1, …, cal Fd is usually measured by the sum or the product of the sizes |cal Fi|. These two natural size functions have different extremal structures, and different techniques have been developed for them. We introduce the cross-degree of a size function and show that the potential extremal structures of maximum cross-intersecting families are the same for all size functions of finite cross-degree, including the sum and the product. Our main tool is a strengthened form of the generating family method of Ahlswede--Khachatrian, based on standard generating families and on local modifications of cross-intersecting families, which we call mutations. It treats two completely different types of extremal structures in a unified way and, unlike the original method, needs shifting in only one step.
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BibTeX
@misc{xiang2026sizes,
title = {{Sizes of cross-intersecting families}},
author = {Ziqing Xiang and Huajun Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38328},
eprint = {2610.38328},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38328},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sizes of cross-intersecting families
AU - Ziqing Xiang
AU - Huajun Zhang
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38328
UR - https://arxiv.org/abs/2610.38328
AB - The size of cross-intersecting families cal F_1, …, cal F_d is usually measured by the sum or the product of the sizes |cal F_i|. These two natural size functions have different extremal structures, and different techniques have been developed for them. We introduce the cross-degree of a size function and show that the potential extremal structures of maximum cross-intersecting families are the same for all size functions of finite cross-degree, including the sum and the product. Our main tool is a strengthened form of the generating family method of Ahlswede--Khachatrian, based on standard generating families and on local modifications of cross-intersecting families, which we call mutations. It treats two completely different types of extremal structures in a unified way and, unlike the original method, needs shifting in only one step.
ER -