论文标题
不冗余的超平面盖
Irredundant hyperplane covers
论文作者
论文摘要
We prove that if $G$ is an abelian group and $H_1x_1,\dots,H_{k}x_k$ is an irredundant (minimal) cover of $G$ with cosets, then $$|G:\bigcap_{i=1}^{k}H_{i}|=2^{O(k)}.$$ This bound is the best possible up to the constant hidden in the $ o（\ cdot）$符号，它解决了Pyber（1996）和Szegedy（2007）的猜想。 我们进一步表明，如果$ g $是某些大质量$ p $的基本$ p $ - 组，而$ h_1，\ dots，h_k $是一系列具有许多重复的超平面，那么上面的界限可以得到改善。 As a consequence, we establish a substantial strengthening of the recently solved Alon-Jaeger-Tarsi conjecture: there exists $α>0$ such that for every invertible matrix $M\in\mathbb{F}_p^{n\times n}$ and any set of at most $p^α$ forbidden coordinates, one can find a vector $ x \ in \ mathbb {f} _p^{n} $，使得$ x $和$ mx $都不具有禁止的坐标。
We prove that if $G$ is an abelian group and $H_1x_1,\dots,H_{k}x_k$ is an irredundant (minimal) cover of $G$ with cosets, then $$|G:\bigcap_{i=1}^{k}H_{i}|=2^{O(k)}.$$ This bound is the best possible up to the constant hidden in the $O(\cdot)$ notation, and it resolves conjectures of Pyber (1996) and Szegedy (2007). We further show that if $G$ is an elementary $p$-group for some large prime $p$, and $H_1,\dots,H_k$ is a sequence of hyperplanes with many repetitions, then the bound above can be improved. As a consequence, we establish a substantial strengthening of the recently solved Alon-Jaeger-Tarsi conjecture: there exists $α>0$ such that for every invertible matrix $M\in\mathbb{F}_p^{n\times n}$ and any set of at most $p^α$ forbidden coordinates, one can find a vector $x\in\mathbb{F}_p^{n}$ such that neither $x$ nor $Mx$ have a forbidden coordinate.
