# Behavior of generalized hypergraph Ramsey numbers

Denote by $$F^{(t)}(n, \alpha)$$ the largest integer for which it is possible to split the $$t$$-tuples of a set $$S$$ of $$n$$ elements into $$2$$ classes so that for every $$X \subset S$$ with $$\vert X\vert \geq F^{(t)}(n, \alpha)$$, each class contains more than $$\alpha \binom{\vert X\vert}{t}$$ $$t$$-tuples of $$X$$. Note that $$F^{(t)}(n,0)$$ is just the usual Ramsey function $$r_t(n,n)$$. It is easy to show that for every $$0 \leq \alpha \leq 1/2$$,

$$\displaystyle c(\alpha) \log n < F^{(2)} (n,\alpha) < c'(\alpha) \log n.$$

It is conjectured [1] that for $$t=2$$,

$$\displaystyle F^{(2)}(n,\alpha) \sim c \log n$$

for an appropriate $$c$$.

As Erdös says in [1], the situation for $$t \geq 3$$ is much more mysterious. It is well-known[1] that if $$\alpha$$ is sufficiently close to $$1/2$$, then

$$\displaystyle c_t(\alpha) (\log n)^{1/(t-1)} < F^{(t)} (n, \alpha) < c'_t(\alpha) (\log n)^{1/(t-1)} .$$

On the other hand, since $$F^{(t)}(n,0)$$ is just the usual Ramsey function, then the old conjecture of Erdös, Hajnal, Rado[2] would imply

$$\displaystyle c_1 \log _{t-1} n < F^{(t)} (n,0) < c_2 \log_{t-1} n.$$

Thus, assuming this conjecture holds, as $$\alpha$$ increases from 0 to $$1/2$$, $$F^{(t)}(n, \alpha)$$ increases from $$\log_{t-1} n$$ to $$(\log n)^{1/(t-1)}$$.

# Problem (\$ 500)

Does the change in $$F^{(t)} (n,\alpha)$$ occur continuously, or are there jumps?

Erdös suspected there might only be one jump, this occurring at 0.

Bibliography
1 P. Erdös, Problems and results on graphs and hypergraphs: similarities and differences, Mathematics of Ramsey theory, Algorithms Combin., 5, (J. Nešetril and V. Rödl, eds.), 12-28, Springer, Berlin, 1990.

2 P. Erdös, A. Hajnal and R. Rado. Partition relations for cardinal numbers, Acta Math. cad. Sci. Hungar. 16 (1965), 93-196.