Abstract
A two-coloring of the verticesX of the hypergraphH=(X, ε) by red and blue hasdiscrepancy d ifd is the largest difference between the number of red and blue points in any edge. A two-coloring is an equipartition ofH if it has discrepancy 0, i.e., every edge is exactly half red and half blue. Letf(n) be the fewest number of edges in ann-uniform hypergraph (all edges have sizen) having positive discrepancy. Erdős and Sós asked: isf(n) unbounded? We answer this question in the affirmative and show that there exist constantsc 1 andc 2 such that
where snd(x) is the least positive integer that does not dividex.
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The first two authors were supported in part by NSF under grant DMS 8406100; the third one was supported in part by NSF under grant DCR 8421341.
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Alon, N., Kleitman, D.J., Pomerance, C. et al. The smallestn-uniform hypergraph with positive discrepancy. Combinatorica 7, 151–160 (1987). https://doi.org/10.1007/BF02579446
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DOI: https://doi.org/10.1007/BF02579446