事件镇问题的超饱和构造
Constructions for supersaturation of eventown problems
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中文总结 AI 辅助
研究事件镇超饱和问题，给出极值构造，扩展奥尼尔猜想中\(s\)的范围，还用对称设计构造证明当\(k\)为偶数且\(4k - 1\)是素数幂时相关结论，确定特定集合中偶大小子集的奇大小两两交集数量。
中文摘要 AI 辅助
本文研究事件镇的超饱和问题。给定一个\(n\)元集的子集族\(\mathcal{A}\)，用\(op(\mathcal{A})\)表示\(\mathcal{A}\)中满足\(|A\cap B|\)为奇数的不同对\(A,B\)的数量。给出了极值事件镇构造，表明对于固定的\(s\leq2^{\lfloor \frac{n}{2} \rfloor}-2\)，存在一个\(n\)元集的\(2^{\lfloor\frac{n}{2}\rfloor}+s\)个偶大小子集的集合，其奇大小的两两交集恰好有\(s\cdot 2^{\lfloor \frac{n}{2} \rfloor-1}\)个。这扩展了奥尼尔猜想中\(s\)的范围。还利用对称设计给出一种构造，证明当\(k\)为偶数且\(4k - 1\)是素数幂时，存在一个\(4k - 1\)元集\(\mathcal{A}_s\)的\(2^{\lfloor\frac{4k-1}{2}\rfloor}+s\)个偶大小子集的集合，\(op(\mathcal{A}_s)=s \cdot 2^{{\lfloor\frac{4k-1}{2}\rfloor}-1}\)，\(1\leq s\leq4k-1\)。
英文摘要
In this paper, we study the supersaturation problems of eventown. Given a family $\mathcal{A}$ of subsets of an $n$ element set, let op$(\mathcal{A})$ denote the number of distinct pairs $A,B\in \mathcal{A}$ for which $|A\cap B|$ is odd. We give extremal eventown constructions and show that for fixed $s\le2^{\lfloor \frac{n}{2} \rfloor}-2$, there exists a collection of $2^{\lfloor\frac{n}{2}\rfloor}+s$ even-sized subsets of an $n$ element set that contains exactly $s\cdot 2^{\lfloor \frac{n}{2} \rfloor-1}$ pairwise intersections of odd size. This extends the range of $s$ in a conjecture proposed by O'Neill from $2^{\lfloor \frac{n}{2} \rfloor}-2^{\lfloor \frac{n}{4} \rfloor}$ to $2^{\lfloor \frac{n}{2} \rfloor}-2$. We also give a construction using symmetric designs to prove that when $k$ is even and $4k-1$ is a prime power, there exists a collection of $2^{\lfloor\frac{4k-1}{2}\rfloor}+s$ even-sized subsets of a $4k-1$ element set $\mathcal{A}_s$ with $op(\mathcal{A}_s)=s \cdot 2^{{\lfloor\frac{4k-1}{2}\rfloor}-1}$, $1\leq s\leq4k-1$.
