Oddtown bounds for composite moduli
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In the composite-modulus Oddtown problem, how large can a family of subsets of an n-element set be when individual set sizes are nonzero modulo ℓ but pairwise intersection sizes are zero modulo ℓ? The sharp prime-power bound is n; the question asks what happens when ℓ has at least two distinct prime factors.
Why it matters and what counts as progress
Why it matters: Composite moduli can behave differently from prime powers in modular set-intersection bounds, even for the first case ℓ=6.
What counts as progress: Construct families exceeding n, match the leading upper coefficient ω(ℓ), or determine lower-order terms for a fixed composite modulus.
How the problem developed
- 1992 · Open problem [1]
László Babai and Péter Frankl Asked whether the n-set upper bound for prime-power Oddtowns also holds for composite moduli that are not prime powers.
- 30 Aug 2025 · Lower bound [2]
Boris Bukh, Ting-Wei Chao and Zeyu Zheng For square-free composite moduli with ω distinct prime factors, constructed families of size ωn−o(n), disproving a universal n bound and establishing the leading asymptotic term.
- 6 Aug 2026 · Upper bound [3]
Yuhao Zhao Improved the upper bound for composite-modulus Oddtowns, obtaining a superlogarithmic saving below ωn for each fixed composite modulus.
AI contribution
The Oddtown problem modulo a composite number
For every fixed modulus ℓ with at least two distinct prime factors, the AI-origin construction gives fℓ(n)≥ω(ℓ)n−o(n) through projective-hyperplane complements and low-rank correction blocks. This refutes fℓ(n)≤n outside prime powers.
Sources
- László Babai and Péter Frankl, Linear Algebra Methods in Combinatorics (1992/2022 edition)
- Boris Bukh, Ting-Wei Chao and Zeyu Zheng, The Oddtown problem modulo a composite number (2025; revised 2026)
- Yuhao Zhao, A superlogarithmic saving for Oddtown modulo composite numbers (2026)
- Bukh, Chao and Zheng, composite Oddtown bounds
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