OPEN
Let $N\geq 1$. What is the largest $t$ such that there are $A_1,\ldots,A_t\subseteq \{1,\ldots,N\}$ with $A_i\cap A_j$ a non-empty arithmetic progression for all $i\neq j$?
Simonovits and Sós
[SiSo81] have shown that $t\ll N^2$. It is possible that the maximal $t$ is achieved when we take the $A_i$ to be all arithmetic progressions in $\{1,\ldots,N\}$ containing some fixed element.
If we drop the non-empty requirement then Simonovits, Sós, and Graham [SiSoGr80] have shown that
\[t\leq \binom{N}{3}+\binom{N}{2}+\binom{N}{1}+1\]
and this is best possible.
