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We prove that for any constant k and any < 1, there exist bimatrix win-lose games for which every-WSNE requires supports of cardinality greater than k. To do this, we provide a graph-theoretic characterization of win-lose games that possess-WSNE with constant cardinality supports. We then apply a result in additive number theory of Haight  to construct win-lose games that do not satisfy the requirements of the characterization. These constructions disprove graph theoretic conjectures of Daskalakis, Mehta and Papadimitriou  and Myers .fatcat:6ihnxpirwfc6fhdxwimrnai4ry