This question was inspired by an attempt to compute the probability of word clustering, but I think it makes sense phrased independently of that.
Let $c$ be a positive integer. Suppose I choose $b$ integers $X_1,\dots,X_b$, randomly and uniformly from the range $[1,c]$ (so the $X_i$ are iid variables with joint distribution the uniform one on the discrete set $\{1,\dots,c\}$). For each integer $a<b$, let $D^{a,b,c}$ be the minimum diameter of a set $\{X_{i_1},\dots,X_{i_a}\}$. So, $D^{1,b,c}=0$ identically.
Is the distribution of $D^{a,b,c} $ known? Or, is there an estimate of that distribution
Motivation: suppose a document has $c$ sentences, and $b$ of them are about some specific topic. We can think of this as corresponding to a sample $x_1,\dots,x_b$. Suppose a surprising number of those sentences are close to each other (i.e., the sample $D^{a,b,c}$ is small for some $a$ large relative to $b$). I'd like to have a direct formula for $P(D^{a,b,c}<t)$, rather than estimating the probability programatically.