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arXiv 1902.10258math.CO
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超图二部Turán问题的新界

New bounds for a hypergraph Bipartite Turán problem

Beka Ergemlidze, Tao Jiang, Abhishek Methuku

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摘要：

令$t$为整数，使得$t\geq 2$。令$K_{2,t}^{(3)}$表示由$2t$三元组$\{a,x_i,y_i\}$、$\{b,x_i,y_i\}$组成的三元系，其中元素$a, b, x_1, x_2, \ldots, x_t,$ $y_1, y_2, \ldots, y_t$都是不同的。令$ex(n,K_{2,t}^{(3)})$表示不包含$K_{2,t}^{(3)}$的$n$元素上的三元系的最大大小。这个函数由Mubayi和Verstraëte研究，其中$t=2$的特殊情况是Erdés的问题，由不同的作者研究。Mubayi和Verstraëte证明了$ex(n,K_{2,t}^{(3)})<t^4\binom{n}{2}$和对无穷多个$n$，$ex(n,K_{2,t}^{(3)})\geq \frac{2t-1}{3} \binom{n}{2}$。这些界限与一个标准的论点一起表明，$g(t):=\lim_{n\to \infty} ex(n,K_{2,t}^{(3)})/\binom{n}{2}$存在，并且\[\frac{2t-1}{3}\leq g(t)\leq t^4.\]解决问题的Mubayi和Verstraëte的增长率的$g(t)$，我们证明，作为$t \to \infty$，\[g(t) = Θ(t^{1+o(1)}).\]

英文摘要：

Let $t$ be an integer such that $t\geq 2$. Let $K_{2,t}^{(3)}$ denote the triple system consisting of the $2t$ triples $\{a,x_i,y_i\}$, $\{b,x_i,y_i\}$ for $1 \le i \le t$, where the elements $a, b, x_1, x_2, \ldots, x_t,$ $y_1, y_2, \ldots, y_t$ are all distinct. Let $ex(n,K_{2,t}^{(3)})$ denote the maximum size of a triple system on $n$ elements that does not contain $K_{2,t}^{(3)}$. This function was studied by Mubayi and Verstraëte, where the special case $t=2$ was a problem of Erdős that was studied by various authors. Mubayi and Verstraëte proved that $ex(n,K_{2,t}^{(3)})<t^4\binom{n}{2}$ and that for infinitely many $n$, $ex(n,K_{2,t}^{(3)})\geq \frac{2t-1}{3} \binom{n}{2}$. These bounds together with a standard argument show that $g(t):=\lim_{n\to \infty} ex(n,K_{2,t}^{(3)})/\binom{n}{2}$ exists and that \[\frac{2t-1}{3}\leq g(t)\leq t^4.\] Addressing the question of Mubayi and Verstraëte on the growth rate of $g(t)$, we prove that as $t \to \infty$, \[g(t) = Θ(t^{1+o(1)}).\]

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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