通过生成器切换实现Eventown的超饱和

中文摘要 AI 辅助

Eventown族是指由[n]中偶大小子集构成的族，其中任意两个不同成员的交集大小均为偶数。Berlekamp和Graver的经典定理表明，这类族的最大规模为2^{⌊n/2⌋}。Eventown的超饱和问题是指，当超过该极值界时，必须出现多少个交集为奇数的子集对。对于[n]中偶大小子集构成的族F，记e(F)为交集大小为奇数的无序对的数量。O'Neill猜想，若|F|=2^{⌊n/2⌋}+s，其中1≤s≤2^{⌊n/2⌋}-2^{⌊n/4⌋}，则e(F)≥s·2^{⌊n/2⌋-1}。此前，该猜想仅在s=1、2，以及n足够大且s≤2^{⌊n/8⌋}/n时成立。本文证明了1≤s≤2^{⌊n/2⌋}/26时的猜想界，将已知范围扩展到极值Eventown大小的固定正比例，且该范围中界是紧的。作为进一步结果，本文推导了对任意超额s均成立的下界，在另一范围改进了先前的估计；还建立了极值大小族的稳定性和移除结果，当e(F)<2^{⌊n/2⌋-1}时，这类族接近极值Eventown族，且删除少量成员即可使其成为Eventown。

英文摘要

An eventown family is a family of even-sized subsets of $[n]$ in which every two distinct members have an even-sized intersection. A classical theorem of Berlekamp and Graver shows that the maximum size of such a family is $2^{\lfloor n/2\rfloor}$. The supersaturation problem for eventown asks how many odd-intersection pairs must occur when this extremal bound is exceeded. For a family $\mathcal F$ of even-sized subsets of $[n]$, let $e(\mathcal F)$ denote the number of unordered pairs whose intersection size is odd. O'Neill conjectured that if $|\mathcal F|=2^{\lfloor n/2\rfloor}+s$, then $e(\mathcal F)\ge s\,2^{\lfloor n/2\rfloor-1}$ for \[ 1\le s\le 2^{\lfloor n/2\rfloor}-2^{\lfloor n/4\rfloor}. \] Previously, the conjecture was known for $s=1,2$, and, for $s\le 2^{\lfloor n/8\rfloor}/n$ with $n$ sufficiently large. We prove the conjectured bound for \[ 1\le s\le \frac{2^{\lfloor n/2\rfloor}}{26}, \] extending the known range to a fixed positive proportion of the extremal eventown size. The bound is sharp throughout this range. As further consequences, we derive a lower bound valid for arbitrary excess $s$, which improves the previously known estimate in an additional range. We also establish stability and removal results for families of extremal size satisfying $e(\mathcal F)<2^{\lfloor n/2\rfloor-1}$, showing that such a family is close to an extremal eventown family and can be made eventown by deleting a small number of its members.

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12 pages