Seminar on Discrete Mathematics: Refined estimates on the clique number of generalized Paley graphs

：叶智恺（加拿大英属哥伦比亚大学）
：2023-08-31 14:30
：海韵园数理大楼686会议室

报 告 人：叶智恺（加拿大英属哥伦比亚大学）

时  间：2023年8月31日14:30

地  点：海韵园数理大楼686会议室

内容摘要:

Let $d \geq 2$ and let $q$ be a prime power such that $q \equiv 1 \pmod {2d}$. The $d$-Paley graph of order $q$ is the graph with is the graph with the vertex set being the finite field $\mathbb{F}_q$, where two vertices are adjacent if and only if their difference is a $d$-th power in $\mathbb{F}_q^*$. In this talk, we show that the clique number of the $d$-Paley graph of order $q$ is at most $\sqrt{q/d}+O(\sqrt{q/p})$, where $q$ is an odd power of a prime $p$. This significantly improves the best-known generic upper bound $\sqrt{q}-o(p)$ and matches with the bound $\sqrt{p/d}+O(1)$ for primes $p$ in a recent breakthrough work of Hanson and Petridis. Moreover, our new bound is asymptotically sharp for an infinite family of graphs, which leads to the further discovery of the first nontrivial instance of families of generalized Paley graphs where the clique number can be explicitly determined.

个人简介：

叶智恺，目前在英属哥伦比亚大学读博, 2019年6月本科毕业于香港科技大学。主要研究方向包括算术组合, 极值组合，代数图论, 和解析数论。已在Finite Fields Appl., J. Algebraic Combin., J. Combin. Theory Ser. A, Mathematika等杂志发表学术论文10篇。

 

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