Given $s \ge k\ge 3$, let $h^{(k)}(s)$ be the minimum $t$ such that there exist arbitrarily large $k$-uniform hypergraphs $H$ whose independence number is at most polylogarithmic in the number of vertices and in which every $s$ vertices span at most $t$ edges. Erd\H os and Hajnal conjectured (1972) that $h^{(k)}(s)$ can be calculated precisely using a recursive formula and Erd\H os offered \$500 for a proof of this. For $k=3$ this has been settled for many values of $s$ including powers of three but it was not known for any $k\geq 4$ and $s\geq k+2$. Here we settle the conjecture for all $s \ge k \ge 4$. We also answer a question of Bhatt and Rodl by constructing, for each $k \ge 4$, a quasirandom sequence of $k$-uniform hypergraphs with positive density and upper density at most $k!/(k^k-k)$. This result is sharp.

中文翻译：

---

拉姆齐理论中的多项式到指数过渡

给定$ s \ ge k \ ge 3 $，令$ h ^ {{（k）}（s）$为最小值$ t $，以便存在任意大的$ k $一致超图$ H $，其独立数为顶点数量最多的多对数形式，并且每个$ s $顶点最多跨越$ t $个边。Erd \ H os和Hajnal猜想（1972年），可以使用递归公式精确地计算$ h ^ {（k）}（s）$，而Erd \ H os提供了$ 500作为证明。对于$ k = 3 $，已经为$ s $的许多值（包括3的幂）进行了结算，但是对于任何$ k \ geq 4 $和$ s \ geq k + 2 $都不知道。在这里，我们解决所有$ s \ ge k \ ge 4 $的猜想。我们还通过为每个$ k \ ge 4 $构造一个$ k $统一超图的准随机序列，其正密度和上限密度最大为$ k！/（k ^ kk）$，来回答Bhatt和Rodl的问题。这个结果很明显。