论文标题
在由笛卡尔产品确定的方向和广义Paley图的集团数量
On the directions determined by Cartesian products and the clique number of generalized Paley graphs
论文作者
论文摘要
众所周知，笛卡尔产品$ a \ times b \ subset ag（2，p）$至少是$ | a || b | - \ min \ {| a |，| b | \} + 2 $，提供$ p $是prime和$ | a || b | <p $。这意味着在$ \ mathbb {f} _p $上的Paley图的集团数字上最著名的上限。在本文中，我们将此结果扩展到$ ag（2，q）$，其中$ q $是主要功率。我们还对$ \ mathbb {f} _q $的总体数量的集团数量进行了改进的上限。特别是，对于立方paley图，我们将琐碎的上限$ \ sqrt {q} $提高到$ 0.769 \ sqrt {q}+1 $。通常，作为我们的主要结果的应用，对于任何正函数$ h $，以便$ h（x）= o（x）$（$ x \ to \ infty $），我们改善了几乎所有非squares $ q $ q $ q $ q $ q $ q $ q $。
It is known that the number of directions formed by a Cartesian product $A \times B \subset AG(2,p)$ is at least $|A||B| - \min\{|A|,|B|\} + 2$, provided $p$ is prime and $|A||B|<p$. This implies the best known upper bound on the clique number of the Paley graph over $\mathbb{F}_p$. In this paper, we extend this result to $AG(2,q)$, where $q$ is a prime power. We also give improved upper bounds on the clique number of generalized Paley graphs over $\mathbb{F}_q$. In particular, for a cubic Paley graph, we improve the trivial upper bound $\sqrt{q}$ to $0.769\sqrt{q}+1$. In general, as an application of our key result on the number of directions, for any positive function $h$ such that $h(x)=o(x)$ as $x \to \infty$, we improve the trivial upper bound $\sqrt{q}$ to $\sqrt{q}-h(p)$ for almost all non-squares $q$.
