Lin, Chenying and Zemor, Gilles (2026) Kneser's theorem for codes and ℓ-divisible set families. FINITE FIELDS AND THEIR APPLICATIONS, 111: 102783. ISSN 1071-5797, 1090-2465
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A k-wise B-divisible set family is a collection F of subsets of {1,. .., n} such that any intersection of k sets in F has cardinality divisible by B. If k = B = 2, it is well-known that F <= 2tn/21. We generalise this by proving that F <= 2tn/p1 if k = B = p, for any prime number p. For arbitrary values of B, we prove that 4B2-wise B-divisible set families F satisfy F <= 2tn/& ell;1 and that the only families achieving the upper bound are atomic, meaning that they consist of all the unions of disjoint subsets of size B. This improves upon a recent result by Gishboliner, Sudakov and Timon, that arrived at the same conclusion for k-wise Bdivisible families, with values of k that behave exponentially in B. Our techniques rely heavily upon a coding-theory analogue of Kneser's Theorem from additive combinatorics. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | Set systems; Coding theory; Code products; Additive combinatorics |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 07 May 2026 06:54 |
| Last Modified: | 07 May 2026 06:54 |
| URI: | https://pred.uni-regensburg.de/id/eprint/66427 |
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