Problem

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Tags: combinatorics unsolved, combinatorics



There are $ n$ voters and $ m$ candidates. Every voter makes a certain arrangement list of all candidates (there is one person in every place $ 1,2,...m$) and votes for the first $ k$ people in his/her list. The candidates with most votes are selected and say them winners. A poll profile is all of this $ n$ lists. If $ a$ is a candidate, $ R$ and $ R'$ are two poll profiles. $ R'$ is $ a-good$ for $ R$ if and only if for every voter; the people which in a worse position than $ a$ in $ R$ is also in a worse position than $ a$ in $ R'$. We say positive integer $ k$ is monotone if and only if for every $ R$ poll profile and every winner $ a$ for $ R$ poll profile is also a winner for all $ a-good$ $ R'$ poll profiles. Prove that $ k$ is monotone if and only if $ k>\frac{m(n-1)}{n}$.