片秩法及其應用

Abstract

片秩法 (Slice Rank) 是 Croot, Lev 以及 Pach 於2016年提出的一個新的組合數學工具，許多極值組合學當中的問題都透過了這個方法有了新的進展。在這篇論文中，我們會介紹片秩法及其應用，同時我們也會介紹劃分秩法 (Partition Rank) 及其應用。最後我們會利用片秩法以及隨機圖的定理證明直角移除定理。

Slice rank methods are new combinatorial tools introduced by Croot, Lev, and Pach in 2016. Many problems in extremal combinatorics are improved by applying the slice rank methods. In this thesis, we'll introduce the slice rank methods and their applications. Moreover, the partition rank and one of its applications are also introduced. Finally, we use slice rank methods and a random graph theorem to prove the right angles removal lemma.