Problem 161 — Visual explainer (undergrad)
==========================================

Problem statement
---------------
Let $\alpha\in[0,1/2)$ and $n,t\geq 1$. Let $F^{(t)}(n,\alpha)$ be the smallest $m$ 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?

Picture
-------
Graph picture:

o---o
|\  |
| \ |
o---o    K4 is the complete graph on 4 vertices (all connections)

Question: what to look at
------------------------
- Restate the problem as: given the constraint, what asymptotic bound or
  structure must follow?
- Use the comments as benchmarks (best known bounds / constructions).

Context
-------
Area hint: infinite graph theory / Ramsey-type decompositions.

Benchmarks / known results (from comments)
----------------------------------------
- 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: '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 $t>3$.