Towards odd-sunflowers: temperate families and lightnings

TL;DR

This paper introduces the concept of temperate families in the hypercube, characterizes their maximum sizes, and classifies extremal families, extending to intersecting cases and conjecturing for even dimensions.

Contribution

It defines temperate families, proves maximum size is achieved by middle layers, and classifies extremal families, including intersecting cases for odd dimensions.

Findings

Maximum size of temperate families is attained by middle layers.

Middle $t+1$ layers maximize families with bounded subset containment.

Classified all maximum extremal families, including intersecting cases for odd $n$.

Abstract

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 $n$.