# 2014-11 Subsets of a countably infinite set

Prove or disprove that every uncountable collection of subsets of a countably infinite set must have two members whose intersection has at least 2014 elements.

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Consider all non-empty subsets $$S_1,S_2,\ldots,S_{2^n-1}$$ of $$\{1,2,3,\ldots,n\}$$. Let $$A=(a_{ij})$$ be a $$(2^n-1)\times(2^n-1)$$ matrix such that $a_{ij}=\begin{cases}1 & \text{if }S_i\cap S_j\ne \emptyset,\\0&\text{otherwise.}\end{cases}$ What is $$\lvert\det A\rvert$$?