- Cosmin Pohoata, A combinatorial proof of Lindstr”om’s theorem on unions of subsets of a finite set. (Link to the PDF file)
In this note, we give a combinatorial proof of Bernt Lindstr\”om’s theorem that if A_{1}, A_{2},…, A_{n+1} are nonempty subsets of an n-element set, then we can find two disjoint and nonempty groups of indices \left\{i_{1},\ i_{2},\ \ldots,\ i_{k}\right\} and \left\{j_{1},\ j_{2},\ \ldots,\ j_{m}\right\} such that
A_{i_{1}} \cup A_{i_{2}} \cup \ldots \cup A_{i_{k}} = A_{j_{1}} \cup A_{j_{2}} \cup \ldots \cup A_{j_{m}}.
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