论文标题
在由载offine galois平面中的笛卡尔产品确定的方向上
On the directions determined by a Cartesian product in an affine Galois plane
论文作者
论文摘要
我们证明，一组$ a \ times b \ subset ag（2，p）$中包含的方向数，其中$ p $是prime，至少为$ | a | a || b | - \ min \ {| a |，| b | \} + 2 $。这里$ a $和$ b $是$ gf（p）$的子集，至少两个元素和$ | a || b | <P $。对于无限类别的示例，这种绑定是紧密的。我们的主要工具是将Rédei多项式与Szőnyi扩展使用。作为我们主要结果的应用，我们获得了佩奇图的集团数量的上限，与Hanson和Petridis最近获得的最佳界限匹配。
We prove that the number of directions contained in a set of the form $A \times B \subset AG(2,p)$, where $p$ is prime, is at least $|A||B| - \min\{|A|,|B|\} + 2$. Here $A$ and $B$ are subsets of $GF(p)$ each with at least two elements and $|A||B| <p$. This bound is tight for an infinite class of examples. Our main tool is the use of the Rédei polynomial with Szőnyi's extension. As an application of our main result, we obtain an upper bound on the clique number of a Paley graph, matching the current best bound obtained recently by Hanson and Petridis.
