Towards odd-sunflowers: temperate families and lightnings

Motivated by odd-sunflowers, introduced recently by Frankl, Pach, and P{\'a}lv{\"o}lgyi, we initiate the study of temperate families: a family $\mathcal{F} \subseteq \mathcal{P}([n])$ is said to be \emph{temperate} if each $A \in \mathcal{F}$ contains at most $|A|$ elements of $\mathcal{F}$ as a proper subset. We show that the maximum size of a temperate family is attained by the middle two layers of the hypercube $\{0,1\}^n$. As a more general result, we obtain that the middle $t+1$ layers of the hypercube maximise the size of a family $\mathcal{F}$ such that each $A \in \mathcal{F}$ contains at most $\sum_{j=1}^t \binom{|A|}{j}$ elements of $\mathcal{F}$ as a proper subset. Moreover, we classify all such families consisting of the maximum number of sets. In the case of intersecting temperate families, we find the maximum size and classify all intersecting temperate families consisting of the maximum number of sets for odd $n$. We also conjecture the maximum size for even $n$.