Consider a set of $n$ vertices of common degree $p$. What is the probability that some subset of $x$ vertices from $n$ share $q$ mutual connections within that group of size $x$?
i.e. If we have 588 vertices, all of degree 46, what is the probability that, if we take a subset of 10 vertices, they all share at least 6 connections with elements of that 10 vertex set?
So far, I've considered that there are $\frac{n*p}{2}$ edges in the graph, of which there are $\frac{(n)(n-1)}{2}$ spaces in which they might be placed (the number of edges on a complete graph), so there would be $$\binom{\frac{(n)(n-1)}{2}}{\frac{np}{2}}$$ ways which one could choose to place these edges. Similarly, there are $\binom{n}{x}$ ways to choose $x$ vertices from a set of $n$ vertices, and $\sum_{i=q}^{x-1} \binom{p}{i}$ ways to choose at least $q$ degrees per vertex for each vertex in the set of size $x$. However, this is where my rather limited knowledge of combinatorics and probabilities breaks down, and quite frankly, I'm not at all sure how to combine these things to form an answer (or if these even are the terms one would require to do so). I know the problem is somewhat Ramseyian in nature, but it still feels (at least, to my untrained eyes) distinct enough that an explicit, albeit likely messy, answer would exist.
Thanks in advance for the time and help!