## Problem of the week

### 2017-04 More than a half

Prove (or disprove) that exactly one of the following is true for every subset $$A$$ of $$\{ (i,j): i,j\in\{1,2,\ldots,n\}, i\neq j\}$$. (i) There exists a sequence of distinct integers $$i_1,i_2,\ldots,i_k\in \{1,2,\ldots,n\}$$ for some integer $$k>1$$ such that $$(i_1,i_2), (i_2,i_3),\ldots,(i_{k-1},i_k), (i_k,i_1)\in A$$. (ii) There exists a collection of finite sets $$A_1,A_2,\ldots,A_n$$ such that for all distinct $$i,j\in\{1,2,\ldots,n\}$$, $$(i,j)\in A$$ if and only if $$\lvert A_i\cap A_j\rvert > \frac12 \lvert A_i\rvert$$ and $$\lvert A_i\cap A_j\rvert \le \frac12 \lvert A_j\rvert$$