论文标题
线性三重系统中的Turan和Ramsey数字
Turan and Ramsey numbers in linear triple systems
论文作者
论文摘要
在本文中，我们在线性三重系统中研究Turán和Ramsey的数字，定义为$ 3 $均匀的超图，其中任何两个三元组最多在一个顶点中相交。 Ruzsa和Szemerédi的著名结果是，对于任何固定的$ c> 0 $和足够大的$ n $，以下Turán-type定理所持。如果$ n $顶点上的线性三重系统至少具有$ cn^2 $边缘，则它包含一个{\ em triangle}：三个成对相交的三个相交的三元组，而没有常见的顶点。在本文中，我们将此结果从三角形扩展到其他三重系统，称为{\ em $ s $ -configurations}。主要的工具是将诱导的匹配引理从$ aba $ - 帕特恩斯概括到更一般的引理。 我们将$ s $ -configurations稍微概括为{\ em扩展$ s $ -configurations}。为此，我们无法证明相应的Turán型定理，但是我们证明它们具有较弱的Ramsey财产：可以在任何足够大的STORINE三重系统的块中找到它们。使用此情况，我们表明，所有不可避免的配置最多都具有5个块，但可能包含帆$ C_ {15} $的配置（带有块123、345、561和147）的配置是$ t $ -ramsey，对于任何$ t \ geq 1 $。其中最有趣的一个是{\ em Wicket}，$ d_4 $，由三行和两个$ 3 \ times 3 $ point矩阵形成。实际上，在很强的意义上，检票口是$ 1 $ -RAMSEY：除Fano飞机以外的所有Steiner三重系统都必须包含一个检票口。
In this paper we study Turán and Ramsey numbers in linear triple systems, defined as $3$-uniform hypergraphs in which any two triples intersect in at most one vertex. A famous result of Ruzsa and Szemerédi is that for any fixed $c>0$ and large enough $n$ the following Turán-type theorem holds. If a linear triple system on $n$ vertices has at least $cn^2$ edges then it contains a {\em triangle}: three pairwise intersecting triples without a common vertex. In this paper we extend this result from triangles to other triple systems, called {\em $s$-configurations}. The main tool is a generalization of the induced matching lemma from $aba$-patterns to more general ones. We slightly generalize $s$-configurations to {\em extended $s$-configurations}. For these we cannot prove the corresponding Turán-type theorem, but we prove that they have the weaker, Ramsey property: they can be found in any $t$-coloring of the blocks of any sufficiently large Steiner triple system. Using this, we show that all unavoidable configurations with at most 5 blocks, except possibly the ones containing the sail $C_{15}$ (configuration with blocks 123, 345, 561 and 147), are $t$-Ramsey for any $t\geq 1$. The most interesting one among them is the {\em wicket}, $D_4$, formed by three rows and two columns of a $3\times 3$ point matrix. In fact, the wicket is $1$-Ramsey in a very strong sense: all Steiner triple systems except the Fano plane must contain a wicket.
