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According to a result of Walkup, the face vector of any triangulated 4-manifold $X$ with Euler characteristic $\\chi$ satisfies $f_1 \\geq 5f_0 - 15/2 \\chi$, with equality only for $X \\in {\\cal K}(4)$. K\\\"{u}hnel observed that this implies $f_0(f_0 - 11) \\geq -15\\chi$, with equality only for 2-neighborly members of ${\\cal K}(4)$. For $n = 6, 11$ and 15, there are triangulated 4-manifolds with $f_0=n$ and $f_0(f_0 - 11) = -15\\chi$. 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