SIAM Journal on Discrete Mathematics, Volume 34, Issue 1, Page 385-398, January 2020.
A celebrated result of Mantel shows that every graph on n vertices with $\lfloor n^2/4 \rfloor + 1$ edges must contain a triangle. A robust version of this result, due to Rademacher, says that there must, in fact, be at least $\lfloor n/2 \rfloor$ triangles in any such graph. Another strengthening, due to the combined efforts of many authors starting with Erdös, says that any such graph must have an edge which is contained in at least $n/6$ triangles. Following Mubayi, we study the interplay between these two results, that is, between the number of triangles in such graphs and their book number, the largest number of triangles sharing an edge. Among other results, Mubayi showed that for any $1/6 \leq \beta < 1/4$ there is $\gamma > 0$ such that any graph on $n$ vertices with at least $\lfloor n^2/4\rfloor + 1$ edges and book number at most $\beta n$ contains at least $(\gamma -o(1))n^3$ triangles. He also asked for a more precise estimate for $\gamma$ in terms of $\beta$. We make a conjecture about this dependency and prove this conjecture for $\beta = 1/6$ and for $0.2495 \leq \beta < 1/4$, thereby answering Mubayi's question in these ranges.

中文翻译：

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极密度下的书籍与三角形

SIAM离散数学杂志，第34卷，第1期，第385-398页，2020年1月。
Mantel的著名结果表明，在$ \ lfloor n ^ 2/4 \ rfloor + 1 $边的n个顶点上的每个图形都必须包含一个三角形。由于Rademacher的缘故，此结果的可靠版本表明，实际上，任何此类图中必须至少有$ \ lfloor n / 2 \ rfloor $个三角形。由于许多作者从Erdös开始的共同努力，另一项增强功能是说，任何这样的图形都必须具有至少包含在$ n / 6 $三角形中的边。继穆巴伊之后，我们研究了这两个结果之间的相互作用，即，此类图中的三角形数量与其书本数量之间的相互作用，最大的三角形数量共享一条边。在其他结果中，Mubayi显示，对于任何$ 1/6 \ leq \ beta <1/4 $，都有$ \ gamma> 0 $使得$ n $顶点上至少具有$ \ lfloor n ^ 2/4 \ rfloor + 1 $边且书号最多$ \ beta n $的任何图至少包含$（\ gamma -o（1） ）n ^ 3 $个三角形。他还要求以\\ beta $来更精确地估算\\ gamma $。我们对此依赖项做出一个猜想，并针对$ \ beta = 1/6 $和$ 0.2495 \ leq \ beta <1/4 $证明这一猜想，从而回答了Mubayi在这些范围内的问题。