OPEN - $500 Let$\alpha\in[0,1/2)$and$n,t\geq 1$. Let$F^{(t)}(n,\alpha)$be the largest$m$such that we can$2$-colour the edges of the complete$t$-uniform hypergraph on$n$vertices such that if$X\subseteq [n]$with$\lvert X\rvert \geq m$then there are at least$\alpha \binom{\lvert X\rvert}{t}$many$t$-subsets of$X$of each colour. For fixed$n,t$as we change$\alpha$from$0$to$1/2$does$F^{(t)}(n,\alpha)$increase continuously or are there jumps? Only one jump? For$\alpha=0$this is the usual Ramsey function. A conjecture of Erdős, Hajnal, and Rado (see [562]) implies that $F^{(t)}(n,0)\asymp \log_{t-1} n$ and results of Erdős 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$1/2$. Erdős believed there might be just one jump, occcurring at$\alpha=0$. Conlon, Fox, and Sudakov [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].