AI 中文总结
研究具有受限差值的最大斯佩纳族问题，通过将有向差值简化为受限汉明距离渐近解决猜想，对\(L = [s]\)且\(n\geq 2s - 1\)情况用新的齐次多项式方法得到精确答案，给出不同条件下\(L -\)差分族大小的界限。
AI 中文摘要
设\(L\)为正整数的固定集合。若对于\(\mathcal{F}\subseteq 2^{[n]}\)中每对不同的成员\(A,B\)，\(\lvert A\setminus B\rvert\in L\)，则称\(\mathcal{F}\)为\(L -\)差分族。1985年Frankl提出的一个长期猜想断言，每个\(L -\)差分族的大小至多为\(\binom{n}{\vert L\vert}\)。我们对每个固定的\(L\)渐近地解决了这个猜想，并在猜想界限可能紧密的唯一情况下得到了精确答案。若\(L\neq [s]\)且\(n\)很大，每个\(L -\)差分族满足\(\lvert \mathcal{F}\rvert \le \left(\frac{s}{s + 1}+o_L(1)\right)\binom{n}{s}\)；若\(L = [s]\)且\(n\geq 2s - 1\)，则\(\lvert \mathcal{F}\rvert\le\binom{n}{s}\)，仅在\(\binom{[n]}{s}\)和\(\binom{[n]}{n - s}\)时取等号。第一个结果通过将有向差值简化为受限汉明距离得出。对于精确结果，我们开发了一种新的齐次多项式方法，该方法可能具有独立的研究价值。
英文摘要
Let $L$ be a fixed set of positive integers. A family $\mathcal{F}\subseteq 2^{[n]}$ is called $L$-differencing if $\lvert A\setminus B\rvert\in L$ for every ordered pair of distinct members $A,B\in\mathcal{F}$. A longstanding conjecture of Frankl, proposed in 1985, asserts that every $L$-differencing family has size at most $\binom{n}{|L|}$. We resolve this conjecture asymptotically for every fixed $L$, and obtain the exact answer in the only case in which the conjectured bound could be tight. (1) If $L\ne [s]$ and $n$ is large, then every $L$-differencing family satisfies $\lvert \mathcal{F}\rvert \le \left(\frac{s}{s+1}+o_L(1)\right)\binom{n}{s}$. (2) If $L=[s]$ and $n\ge 2s-1$, then $\lvert \mathcal{F}\rvert\le\binom{n}{s}$, with equality only for $\binom{[n]}{s}$ and $\binom{[n]}{n-s}$. The first result follows by reducing directed differences to restricted Hamming distances. For the exact result, we develop a new homogeneous polynomial method, which might be of independent interest.
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