 Modular intersection graph

For a prime number $p$, the vertices are $k$-subsets of $[N]$. An independent set avoids the only off-diagonal intersection congruent to $k$ modulo $p$, so the Frankl-Wilson theorem bounds its size by $\left(\genfrac{}{}{0ex}{}{N}{p-1}\right)$. A clique has constant intersection $p-1$ and has size at most $N$, by the same theorem modulo a prime larger than $k$. This explicit graph gives an asymmetric Ramsey number lower bound and an exponential chromatic number lower bound.