Combining the Theorems of Turán and de Bruijn–Erdős

IF 0.8 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs Pub Date : 2026-04-13 Epub Date: 2026-02-04 DOI:10.1002/jcd.70008

Sayok Chakravarty, Dhruv Mubayi

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引用次数: 0

Abstract

Fix an integer $s \geq 2$ . Let $P$ be a set of $n$ points and let $ℒ$ be a set of lines in a linear space such that no line in $ℒ$ contains more than $(n - 1) / (s - 1)$ points of $P$ . Suppose that for every $s$ -set $S$ in $P$ , there is a pair of points in $S$ that lies in a line from $ℒ$ . We prove that $∣ L ∣ \geq (n - 1) / (s - 1) + s - 1$ for $n$ large, and this is sharp when $n - 1$ is a multiple of $s - 1$ . This generalizes the de Bruijn–Erdős theorem, which is the case $s = 2$ . Our result is proved in the more general setting of linear hypergraphs.

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结合Turán和de Bruijn-Erdős定理

固定一个整数s≥2。设P是n个点的集合设f是线性空间中的一组直线，其中没有直线包含大于(n−1)/（s−1）个P。假设对于P中的每个s -集合s，S中有一对点在一条直线上。我们证明了∣L∣≥（n−1）/ (s−1)+ s−1表示n大，当n - 1是s - 1的倍数时，这个很明显。这推广了de Bruijn-Erdős定理，也就是s = 2的情况。我们的结果在更一般的线性超图集合中得到了证明。

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来源期刊

Journal of Combinatorial Designs 数学-数学

CiteScore

1.60

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： The Journal of Combinatorial Designs is an international journal devoted to the timely publication of the most influential papers in the area of combinatorial design theory. All topics in design theory, and in which design theory has important applications, are covered, including: block designs, t-designs, pairwise balanced designs and group divisible designs Latin squares, quasigroups, and related algebras computational methods in design theory construction methods applications in computer science, experimental design theory, and coding theory graph decompositions, factorizations, and design-theoretic techniques in graph theory and extremal combinatorics finite geometry and its relation with design theory. algebraic aspects of design theory. Researchers and scientists can depend on the Journal of Combinatorial Designs for the most recent developments in this rapidly growing field, and to provide a forum for both theoretical research and applications. All papers appearing in the Journal of Combinatorial Designs are carefully peer refereed.