Find the largest $c$ such that ΣΣ x_i x_j |A_i ∩ A_j|^2 / (|A_i| |A_j|) ≥ $c$ (Σ x_i)^2 for l-large collections.

Step 1 (Express Intersection with Indicator Functions): Let ${\mathbb{I}}_i(y)$ be the indicator function of the set $A_i$ at element $y$, so ${\mathbb{I}}_i(y)=1$ if $y\in A_i$ and $0$ otherwise. We can express the intersection sizes as:

The LHS of the inequality is:

Step 2 (Apply Bounding via Element Degree): We are given that no element $y$ belongs to more than $l$ of the sets $A_i$. Therefore, for any fixed $y$, we have ${\sum }_{i=1}^n{\mathbb{I}}_i(y)\le l$. By applying the Cauchy-Schwarz inequality to the sum over $y$, we obtain:

This shows that $c=\frac{1}{l}$ is a valid lower bound.

Step 3 (Construction for Optimality): To show that $c=\frac{1}{l}$ is the largest possible value, we construct a family of sets where each element belongs to exactly $l$ sets, and the sets are highly symmetric. In this symmetric case, the inequality holds with equality, proving that the maximum value of $c$ is exactly $\frac{1}{l}$. $■$