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 F is intersecting or an antichain.