A Note on $K_{k,k}$-Cross Free Families - Suk, Andrew (The Electronic Journal of Combinatorics, 2008)
We give a short proof that for any fixed integer $k$, the maximum number size of a $K_{k,k}$-cross free family is linear in the size of the groundset. We also give tight bounds on the maximum size of a $K_k$-cross free family in the case when ${\cal F}$ is intersecting or an antichain.