The 1-skeleton of a \(d\)-dimensional polyhedron is abstracted as follows. One considers a graph with vertices some \(d\)-subsets of a base set of \(n\) elements (the ‘facets’), and edges satisfying that for any two vertices \(u,v\) a connecting path exists on which all vertices contain \(u\cap v\). It is shown that the largest possible diameter has a superlinear lower bound \(\Omega(n^2/\log n)\), while both known upper bounds for polyhedra, \(n^{1+\log d}\) by G. Kalai and D. J. Kleitman [Bull. Am. Math. Soc., New Ser. 26, No. 2, 315–316 (1992; Zbl 0751.52006) and \(2^{d-1}n\) by D. G. Larman [Proc. Lond. Math. Soc., III. Ser. 20, 161–178 (1970; Zbl 0199.59301)], still hold.