论文标题
关于Schreier型集合与Turán图的修改之间的关系
On a Relation between Schreier-type Sets and a Modification of Turán Graphs
论文作者
论文摘要
最近，发现了Schreier型集合与Turán图之间的关系。在本说明中，我们给出一个组合证明并获得关系的概括。具体来说，对于$ p，q \ ge 1 $，让$$ \ mathcal {a} _q：= \ {f \ subset \ mathbb {n}：| f | = 1 \ mbox {or} f \ mbox {是算术的进度，有差异} q \} $$和$$ sr（n，p，p，q）\：= \ \ \ \ \ \ \ \ {f \ subset \ subset \ {1，\ ldots，\ ldots，\ ldots，n \} \ Mathcal {a} _q \}。$$我们表明$$ sr（n，p，q）\ = \ t（n+1，pq+1，q），$ t（\ cdot，\ cdot，\ cdot，\ cdot）$是$ n $ vertex图形的边缘数量，该图形是An $ n $ - vertex的图形，该图形是一个Modification tural ofturán的图形。我们还证明$ sr（n，p，q）$是某些序列的部分总和。
Recently, a relation between Schreier-type sets and Turán graphs was discovered. In this note, we give a combinatorial proof and obtain a generalization of the relation. Specifically, for $p, q\ge 1$, let $$\mathcal{A}_q := \{F\subset\mathbb{N}: |F| = 1 \mbox{ or }F\mbox{ is an arithmetic progression with difference } q\}$$ and $$Sr(n, p, q)\ :=\ \#\{F\subset \{1, \ldots, n\}\,:\, p\min F\ge |F|\mbox{ and }F\in \mathcal{A}_q\}.$$ We show that $$Sr(n, p, q) \ =\ T(n+1, pq+1, q),$$ where $T(\cdot, \cdot, \cdot)$ is the number of edges of an $n$-vertex graph that is a modification of Turán graphs. We also prove that $Sr(n,p,q)$ is the partial sum of certain sequences.
