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Infinite number of sets with an intersection property

Source: Romania TST 2013 Test 2 Problem 4

April 26, 2013

analytic geometryalgebrapolynomialcombinatorics proposedcombinatorics

Problem Statement

Let

k

be a positive integer larger than

1

. Build an infinite set

\mathcal{A}

of subsets of

\mathbb{N}

having the following properties:
(a) any

k

distinct sets of

\mathcal{A}

have exactly one common element; (b) any

k+1

distinct sets of

\mathcal{A}

have void intersection.