论文标题
在Prime Power Order的Paley图中数量
On the clique number of Paley graphs of prime power order
论文作者
论文摘要
在添加剂组合学中，找到一个合理的上界的Paley图数量是一个开放的问题。 Hanson和Petridis使用Stepanov方法的最新突破，在Prime Field $ \ Mathbb {F} _p $上定义的Paley图上有改进的上限，其中$ p \ equiv 1 \ pmod 4 $。我们将他们的想法扩展到有限字段$ \ mathbb {f} _q $，其中$ q = p = p^{2s+1} $，用于prime $ p \ equiv 1 \ equiv 1 \ pmod 4 $和非负整数$ s $。我们在$ \ mathbb {f} _ {p^{2s+1}} $上显示了Paley图的集团数字，最多是$ \ min \ bigG（p^s \ bigg \ big \ lceil \ sqrt {\ frac {\ frac {p} {p}} {2}} {2}} {2}} \ sqrt {\ frac {q} {2}}+\ frac {p^s+1} {4} {4}+\ frac {\ sqrt {2p}} {32} {32} p^{s-1} \ bigG）$。
Finding a reasonably good upper bound for the clique number of Paley graphs is an open problem in additive combinatorics. A recent breakthrough by Hanson and Petridis using Stepanov's method gives an improved upper bound on Paley graphs defined on a prime field $\mathbb{F}_p$, where $p \equiv 1 \pmod 4$. We extend their idea to the finite field $\mathbb{F}_q$, where $q=p^{2s+1}$ for a prime $p\equiv 1 \pmod 4$ and a non-negative integer $s$. We show the clique number of the Paley graph over $\mathbb{F}_{p^{2s+1}}$ is at most $\min \bigg(p^s \bigg\lceil \sqrt{\frac{p}{2}} \bigg\rceil, \sqrt{\frac{q}{2}}+\frac{p^s+1}{4}+\frac{\sqrt{2p}}{32}p^{s-1}\bigg)$.
