For \(k\geq1\text{,}\) let \(\mathcal{A}\subseteq \binom{\mathbb{N}}{k}\) be an intersecting family. Does there necessarily exist a finite set \(F\) such that \(\{A\cap F: A\in\mathcal{A}\}\) is intersecting?
Given a set \(A\subseteq\{1,\dots,n\}\) and \(p\in(0,1/2)\text{,}\) let the \(p\)-measure of \(A\) be
Prove that if \(\mathcal{F}\subseteq 2^{[n]}\) is intersecting, then \(\mu_p(\mathcal{F})\leq p\text{.}\)
The mayor of Redundantville has decided that the system for controlling the traffic light at the corner of Main Street and First Avenue is not complicated enough. In order to remedy this, he replaces the current system (a single switch with three positions for the colours green, yellow and red) with a new system that uses \(n\) switches, each of which has three possible positions. The new system has the property that, whenever you change all of the switches to a different position (at the same time), the colour of the light always changes. Prove that there is a single switch whose position completely determines the colour of the light; therefore, the mayor’s efforts to complexify the system were in vain.
