MathDB

Let

n > 2

be an integer and let

\ell \in \{1, 2,\dots, n\}

. A collection

A_1,\dots,A_k

of (not necessarily distinct) subsets of

\{1, 2,\dots, n\}

is called

\ell

-large if

|A_i| \ge \ell

for all

1 \le i \le k

. Find, in terms of

n

and

\ell

, the largest real number

c

such that the inequality

\sum_{i=1}^k\sum_{j=1}^k x_ix_j\frac{|A_i\cap A_j|^2}{|A_i|\cdot|A_j|}\ge c\left(\sum_{i=1}^k x_i\right)^2

holds for all positive integer

k

, all nonnegative real numbers

x_1,x_2,\dots,x_k

, and all

\ell

-large collections

A_1,A_2,\dots,A_k

of subsets of

\{1,2,\dots,n\}

.
Proposed by Titu Andreescu and Gabriel Dospinescu

Problem Statement