تقرير
On the $(k+2,k)$-problem of Brown, Erd\H{o}s and S\'os for $k=5,6,7$
| العنوان: | On the $(k+2,k)$-problem of Brown, Erd\H{o}s and S\'os for $k=5,6,7$ |
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| المؤلفون: | Glock, Stefan, Kim, Jaehoon, Lichev, Lyuben, Pikhurko, Oleg, Sun, Shumin |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Combinatorics |
| الوصف: | Let $f^{(r)}(n;s,k)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph containing no subgraph with $k$ edges and at most $s$ vertices. Brown, Erd\H{o}s and S\'os [New directions in the theory of graphs (Proc. Third Ann Arbor Conf., Univ. Michigan 1971), pp. 53--63, Academic Press 1973] conjectured that the limit $\lim_{n\rightarrow \infty}n^{-2}f^{(3)}(n;k+2,k)$ exists for all $k$. The value of the limit was previously determined for $k=2$ in the original paper of Brown, Erd\H{o}s and S\'os, for $k=3$ by Glock [Bull. Lond. Math. Soc. 51 (2019) 230--236] and for $k=4$ by Glock, Joos, Kim, K\"uhn, Lichev and Pikhurko [arXiv:2209.14177, accepted by Proc. Amer. Math. Soc.] while Delcourt and Postle [arXiv:2210.01105, accepted by Proc. Amer. Math. Soc.] proved the conjecture (without determining the limiting value). In this paper, we determine the value of the limit in the Brown-Erd\H{o}s-S\'os Problem for $k\in \{5,6,7\}$. More generally, we obtain the value of $\lim_{n\rightarrow \infty}n^{-2}f^{(r)}(n;rk-2k+2,k)$ for all $r\geq 3$ and $k\in \{5,6,7\}$. In addition, by combining these new values with recent results of Bennett, Cushman and Dudek [arXiv:2309.00182] we obtain new asymptotic values for several generalised Ramsey numbers. Comment: 44 pages, 3 figures |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2403.04474 |
| رقم الأكسشن: | edsarx.2403.04474 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |