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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2509.21756 |
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| _version_ | 1866911176746598400 |
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| author | Luo, Zilin |
| author_facet | Luo, Zilin |
| contents | Let $K_{2,t}$ denote the complete bipartite graph. For an integer $n\ge 1$, let $ex(n,n,n,K_{2,t})$ be the maximum number of edges in an $n\times n\times n$ tripartite graph (that is, a 3-partite graph with three parts each of size $n$) containing no copy of $K_{2,t}$. In this paper we prove that, for even $t\ge 2$, $$ ex(n,n,n,K_{2,t}) \ge \frac{3\sqrt{t-1}}{\sqrt{2}}\, n^{3/2} + o(n^{3/2}). $$ Combining our construction with earlier work of Tait and Timmons, we obtain $$ \lim\limits_{n\to\infty} \frac{ex(n,n,n,K_{2,t})}{n^{3/2}} = \frac{3\sqrt{t-1}}{\sqrt{2}}, \qquad\text{for integer } t\ge 2. $$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_21756 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Precise Asymptotics of $ex(n,n,n,K_{2,t})$ for even $t$ Luo, Zilin Combinatorics Let $K_{2,t}$ denote the complete bipartite graph. For an integer $n\ge 1$, let $ex(n,n,n,K_{2,t})$ be the maximum number of edges in an $n\times n\times n$ tripartite graph (that is, a 3-partite graph with three parts each of size $n$) containing no copy of $K_{2,t}$. In this paper we prove that, for even $t\ge 2$, $$ ex(n,n,n,K_{2,t}) \ge \frac{3\sqrt{t-1}}{\sqrt{2}}\, n^{3/2} + o(n^{3/2}). $$ Combining our construction with earlier work of Tait and Timmons, we obtain $$ \lim\limits_{n\to\infty} \frac{ex(n,n,n,K_{2,t})}{n^{3/2}} = \frac{3\sqrt{t-1}}{\sqrt{2}}, \qquad\text{for integer } t\ge 2. $$ |
| title | On the Precise Asymptotics of $ex(n,n,n,K_{2,t})$ for even $t$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.21756 |