MathDB

A collection

F

of distinct (not necessarily non-empty) subsets of

X = \{1,2,\ldots,300\}

is lovely if for any three (not necessarily distinct) sets

A

B

and

C

F

at most three out of the following eight sets are non-empty \begin{align*}A \cap B \cap C, \ \ \ \overline{A} \cap B \cap C, \ \ \ A \cap \overline{B} \cap C, \ \ \ A \cap B \cap \overline{C}, \\ \overline{A} \cap \overline{B} \cap C, \ \ \ \overline{A} \cap B \cap \overline {C}, \ \ \ A \cap \overline{B} \cap \overline{C}, \ \ \ \overline{A} \cap \overline{B} \cap \overline{C} \end{align*} where

\overline{S}

denotes the set of all elements of

X

which are not in

S

.
What is the greatest possible number of sets in a lovely collection?

Problem Statement