论文标题
拉姆西的书籍和quasirandomness
Ramsey numbers of books and quasirandomness
论文作者
论文摘要
书形图$ b_n^{（k）} $由$ k_ {k+1} $的$ n $副本组成。已知$ b_n^{（k）} $的Ramsey数字与经典的Ramsey数字有很强的连接。最近，第一作者确定了这些固定$ k $的拉姆齐号码的渐近顺序，从而回答了Erdős，Faudree，Rousseau和Schelp的旧问题。在本文中，我们首先提供了该定理的简单证明。接下来，回答第一作者的问题，我们提供了一个不同的证据，以避免使用Szemerédi的规律性引理，从而对错误术语提供了更严格的控制。最后，我们证明了Nikiforov，Rousseau和Schelp的猜想，证明了这个Ramsey问题的所有极端着色都是Quasirandom。
The book graph $B_n^{(k)}$ consists of $n$ copies of $K_{k+1}$ joined along a common $K_k$. The Ramsey numbers of $B_n^{(k)}$ are known to have strong connections to the classical Ramsey numbers of cliques. Recently, the first author determined the asymptotic order of these Ramsey numbers for fixed $k$, thus answering an old question of Erdős, Faudree, Rousseau, and Schelp. In this paper, we first provide a simpler proof of this theorem. Next, answering a question of the first author, we present a different proof that avoids the use of Szemerédi's regularity lemma, thus providing much tighter control on the error term. Finally, we prove a conjecture of Nikiforov, Rousseau, and Schelp by showing that all extremal colorings for this Ramsey problem are quasirandom.
