On the maximal size of $(a,b)$-town$(\mathrm{m}\mathrm{o}\mathrm{d}k)$ families

A family $\mathcal{F}\subseteq \mathcal{P}(n)$ is an $(a,b)$-town$(\mathrm{m}\mathrm{o}\mathrm{d}k)$ if all sets in it have cardinality $a(\mathrm{m}\mathrm{o}\mathrm{d}k)$ and all pairwise intersections in it have cardinality $b(\mathrm{m}\mathrm{o}\mathrm{d}k)$. For $k=2$ the maximal size of such a family is known for each $a,b$, while for $k=3$ only $b-a\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}3)$ is fully understood. We provide a bound for $k=3$ when $b-a\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}3)$ and $n\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}3)$, which turns out to be tight for infinitely many such $n$. We also give sufficient conditions on the parameters $a,b,k,n$, which result in a better bound than the one from general settings by Ray-Chaudhuri–Wilson, in particular showing that this bound occurs infinitely often in a sense where all of $a,b,n$ can vary for a fixed $k$.