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[For a hypergraph H let β(H) denote the minimal number of edges from H covering V (H). An edge S of H is said to represent fairly (resp. almost fairly) a partition (V1, V2, …, Vm) of V (H) if |S∩Vi|≥|Vi|β(H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\vert S \cap V _{i}\vert \geqslant \left \lfloor \frac{\vert V _{i}\vert } {\beta (H)} \right \rfloor$$ \end{document} (resp. |S∩Vi|≥|Vi|β(H)−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\vert S \cap V _{i}\vert \geqslant \left \lfloor \frac{\vert V _{i}\vert } {\beta (H)} \right \rfloor - 1$$ \end{document}) for all i≤m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i\leqslant m$$ \end{document}. In matroids any partition of V (H) can be represented fairly by some independent set. We look for classes of hypergraphs H in which any partition of V (H) can be represented almost fairly by some edge. We show that this is true when H is the set of independent sets in a path, and conjecture that it is true when H is the set of matchings in Kn, n. We prove that partitions of E(Kn, n) into three sets can be represented almost fairly. The methods of proofs are topological.]
Published: Oct 6, 2017
Keywords: Fair Representation; Hypergraph; Matching Complexity; Borsuk-Ulam Theorem; Matroid Intersection Theorem
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