Problem 2

Let

n

be positive and let

A_1,\ldots,A_{2n+1}

be subsets of

B

. Suppose each

A_i

has exactly

2n

elements, every two distinct

A_i

have exactly one common element, and every element of

B

belongs to at least two

A_i

. For which

n

can one label each element 0 or 1 so that each

A_i

has 0 on exactly

n

elements?

Step 4 of 5: Step 4

Previous step Next step

j-i\in\{-n/2,\ldots,-1,1,\ldots,n/2\}\pmod{2n+1}

Detailed analysis

For even n, index sets cyclically modulo 2n+1 and label (i,j) by 0 when j−i is one of the n residues −n/2,...,−1,1,...,n/2; label all other pairs 1. Each A_i then has exactly n zeros.