For $\alpha=0$ this is the usual Ramsey function.

A conjecture of Erd\H{o}s, Hajnal, and Rado (see [562]) implies that\[ F^{(t)}(n,0)\asymp \log_{t-1} n\]and results of Erd\H{o}s and Spencer imply that\[F^{(t)}(n,\alpha) \gg_\alpha (\log n)^{\frac{1}{t-1}}\]for all $\alpha>0$, and a similar upper bound holds for $\alpha$ close to $1/2$.

Erd\H{o}s said in \cite{Er90b}: 'If I can hazard a guess completely unsupported by evidence, I am afraid that the jump occurs all in one step at $0$. It would be much more interesting if my conjecture would be wrong and perhaps there is some hope for this for $t>3$. I know nothing and offer \$500 to anybody who can clear up this mystery.'

Conlon, Fox, and Sudakov \cite{CFS11} have proved that, for any fixed $\alpha>0$,\[F^{(3)}(n,\alpha) \ll_\alpha \sqrt{\log n}.\]Coupled with the lower bound above, this implies that there is only one jump for fixed $\alpha$ when $t=3$, at $\alpha=0$.

For all $\alpha>0$ it is known that\[F^{(t)}(n,\alpha)\gg_t (\log n)^{c_\alpha}.\]See also [563] for more on the case $t=2$.

This problem is #40 in Ramsey Theory in the graphs problem collection.


References


[CFS11] Conlon, David and Fox, Jacob and Sudakov, Benny, Large almost monochromatic subsets in hypergraphs. Israel J. Math. (2011), 423--432.

[Er90b] Erd\H{o}s, Paul, Problems and results on graphs and hypergraphs: similarities and differences. Mathematics of Ramsey theory (1990), 12-28.