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Partitions.

Source: Romanian TST 1978, Day 4, P3

October 1, 2018
set theorypartitions

Problem Statement

Let p p be a natural number and let two partitions A={A1,A2,...,Ap},B={B1,B2,...Bp} \mathcal{A} =\left\{ A_1,A_2,...,A_p\right\} ,\mathcal{B}=\left\{ B_1,B_2,...B_p\right\} of a finite set M. \mathcal{M} . Knowing that, whenever an element of A \mathcal{A} doesn´t have any elements in common with another of B, \mathcal{B} , it holds that the number of elements of these two is greater than p, p, prove that ∣M∣≥12(1+p2). \big| \mathcal{M}\big|\ge\frac{1}{2}\left( 1+p^2\right) . Can equality hold?
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