Constants for diagonal hypergraph Ramsey Theorem

Question

For integers $n,s$, we write $K_n^{(s)}$ to denote the complete $s$-uniform hypergraph on $n$ vertices.

Given integers $k,s,r$, let $R(K_k^{(s)};r)$ denote the smallest $N$ such that, for every $r$-coloring $C: E(K_N^{(s)}) \to \{1,\dots,r\}$, we can find a monochromatic copy of $K_k^{(s)}$.

It is known that for each $s,r$, that there exists some constant $C=C_{s,r}$ where $$R(K_k^{(s)};r)\le \text{tow}_{s-1}(Ck)$$ for all $k$ (here $\text{tow}_0(x) := x$ and $\text{tow}_{i+1}(x) := 2^{\text{tow}_i(x)}$ for $i\ge 0$). Let $C_{s,r}^*$ be the infinimum of $C$ where such an upper bound holds for all sufficiently large $k$.

It is known that $r\ll C_{2,r}^* \ll r \log r$ for all $r\ge 2$. Meanwhile, we have that $C_{s,r}^*$ is finite for all $s,r$ and know that $C_{s,r}^*>0$ for all $s\ge 2$ and $r\ge 4$.

I was wondering what are the asymptotic bounds we get in general for $C_{s,r}^*$ for fixed $s\ge 3$ and large $r$? Are the lower and upper bounds the same up to polynomial factors, or is there a noticeable gap?

Answer 1

The best source I know for bounds on Ramsey numbers, including hypergraph Ramsey numbers, is Radziszowski's article Small Ramsey Numbers, which constantly gets updated. Since the question didn't contain any sources, I wasn't sure if you were already aware of this one. If the answer is not in here, then the problem is likely open.

Answer 2

Hm, looking at Erdos-Rado Combinatorial theorems on classification of subsets of a given set (Theorem 1), we get that:

$$C_{s,r}\le r\cdot \log_2(r^2 \log_2(r^3\log_2(r^4\dots \log_2(r^s)))))), $$which can alternatively be written as: $$r\cdot \log_2(r^2)\cdot \log_2(2\log_2(r^3))\dots.$$Noting that $\log_2(2x)\ge 2$ for all $x\ge 2$, this upper bound grows exponentially in $s$ for fixed $r$...

Meanwhile, checking the specifics of the stepping up lemma (e.g., https://arxiv.org/pdf/0907.0283.pdf, Theorem 1), it is not hard to see that $C_{s,r}\ge C_{s-1,r}\gg r$.

So for fixed $r$, the bounds differ by a factor of $\exp(s)$, but for fixed $s$, the bounds should differ by a factor of $O_s(\log^{1+o(1)}(r))$.