Based on a conversation with Dan Romik, here is a generalization of Doug's bounds on volume.

Let $H(n,t)$ be the hypergraph of all $t$-tuples of a set with $n$ elements, and let $n > k > t$ be another integer. Suppose each hyperedge $T$ is colored by an i.i.d. random variable $x_T$ that takes values in some measure space $X$. Suppose furthermore that for each subset of $k$-subset $K$, there is some non-trivial symmetric, measurable restriction $R \subset X^{\binom{k}{t}}$ on the colors $x_T$ for $T \subseteq K$. Let $P(n)$ be the probability that all of the restrictions on the coloring of $H(n,t)$ are satisfied.

**Theorem:** For every $m \ge k$,
$$\limsup_{n \to \infty} \frac{\log P(n)}{n^t} \le \frac{\log P(m)}{m^t}.$$

**Corollary:** The limit
$$\alpha = \lim_{n \to \infty} \frac{\log P(n)}{n^t}$$
exists, and one obtains better and better bounds on $\alpha$ by computing $P(m)$ for specific values of $m$, beginning with the case $m=k$ which implies that $\alpha < 0$. In general one obtains $\alpha \in [-\infty,0)$.

Proof. The theorem is a corollary of Rödl's theorem that there exists a packing of blocks of size $k$ which are disjoint on hyperedges of $H(n,t)$, and which cover a fraction of the $t$-tuples that converges to 1 as $n \to \infty$.

**Theorem:** (1) If the condition $R$ contains a cube $I^{\binom{k}{t}}$, where $I \subset X$ is some event with positive measure, then $\alpha > -\infty$. (2) If there is a finite partition $\{I_i\}$ of $X$ such that $R$ is disjoint from each $I_i^{\binom{k}{t}}$, then $\alpha = -\infty$ because $P(n) = 0$ when $n$ is large enough.

For instance, suppose that $X$ is a compact Riemannian manifold with Riemannian measure. Then condition (1) is satisfied if the interior of $R$ contains at least one point on the diagonal. Condition (2) is satisfied if the closure of $R$ is disjoint from the diagonal.

Proof. Case (1) is just the remark that the probability $P(n)$ is at least the probability of landing in $I^{\binom{n}{t}}$. Case (2) follows from Ramsey's theorem.