co.combinatorics - What is the minimum set of combinations C(p,n) required to guarantee q

You may find the following interesting.

I think what you're asking for is the minimum cardinality of a code of length $p$ over the alphabet $Z_n$ (the integers mod $n$, assuming) the possible numbers are $1,2,ldots,n-1$ with covering radius $p-q$. This is in general a very hard NP-complete problem.

Given two vectors $x,y$ in $Z_n^p$, their Hamming distance $d(x,y)$ is defined as the number of coordinates on which they differ. Given a subset $C$ of $Z_n^p$ which corresponds to the collection of lottery selections, the covering radius $R(C)$ is

$$
R(C) = max d(x,C)
$$

where the maximum is taken as $x$ ranges over $Z_n^p$ and where $d(x,C)=min d(x,c)$ with the minimum taken over $c in C.$ When you think about it, you're asking for the minimal cardinality code with covering radius $p-q,$ i.e., with $p-q$ wrong numbers out of $p$.

In general there are bounds on $R(C)$ and the case $n=3$ is of interest in football [soccer] pools. The Golay code is relevant here since it is a perfect ternary code with good covering radius.

There was a nice article MAA monthly years ago entitled something like "football pools, a problem for mathematicians".