Near-sunflowers and focal families

We present some problems and results about variants of sunflowers in families of sets. In particular, we improve an upper bound of the first author, K"orner and Monti on the maximum number of binary vectors of length $n$ so that every four of them are split into two pairs by some coordinate. We also propose a weaker version of the Erd\H{o}s-Rado sunflower conjecture.