论文标题
有限差异的海绵系统
Sperner systems with restricted differences
论文作者
论文摘要
令$ \ Mathcal {f} $为$ [n] $的子集的家族，$ l $为$ [n] $的子集。我们说，如果$ | | a \ setminus b | \ in l $ in l $ in \ Mathcal {f} $。让$ p $为prime，$ q $是$ p $的功率。弗兰克尔（Frankl）首先研究了$ p $ - 模块化$ l $ -differencing sperner Systems，并显示了$ \ sum_ {i = 0}^{| l |} \ binom {n} {i} {i} $的上限的上限。在本文中，我们使用基本$ p $ -ADIC分析和多项式方法获得了$ q $ - 模块化$ l $ d $ ddferencing sperner Systems的新上限，从而大大扩展并改善了现有结果。此外，我们的技术可用于在限制性锤距离距离距离距离距离的超立方体子集上的新上限。该论文的一个亮点是$ Q $二型设置中著名定理的第一个类似物，这导致了几个新的上限，以$ q $ - 模块化$ l $ -y-of-l $ l $ l $ l $ - 更换系统。特别是，我们改善了Felszeghy，Hungeds和Rónyai的结果，并为Babai，Frankl，Kutin和štefankovič提出的问题提供了部分答案。
Let $\mathcal{F}$ be a family of subsets of $[n]$ and $L$ be a subset of $[n]$. We say $\mathcal{F}$ is an $L$-differencing Sperner system if $|A\setminus B|\in L$ for any distinct $A,B\in\mathcal{F}$. Let $p$ be a prime and $q$ be a power of $p$. Frankl first studied $p$-modular $L$-differencing Sperner systems and showed an upper bound of the form $\sum_{i=0}^{|L|}\binom{n}{i}$. In this paper, we obtain new upper bounds on $q$-modular $L$-differencing Sperner systems using elementary $p$-adic analysis and polynomial method, extending and improving existing results substantially. Moreover, our techniques can be used to derive new upper bounds on subsets of the hypercube with restricted Hamming distances. One highlight of the paper is the first analogue of the celebrated Snevily's theorem in the $q$-modular setting, which results in several new upper bounds on $q$-modular $L$-avoiding $L$-intersecting systems. In particular, we improve a result of Felszeghy, Hegedűs, and Rónyai, and give a partial answer to a question posed by Babai, Frankl, Kutin, and Štefankovič.
