  We prove that for any constant $k$ and any $\epsilon<1$, there exist bimatrix
win-lose games for which every $\epsilon$-WSNE requires supports of cardinality
greater than $k$. To do this, we provide a graph-theoretic characterization of
win-lose games that possess $\epsilon$-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.
