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10-29
In this paper, we study the neighbourlicity of the polytope $P_{k n}^2$ constituted by the $k$-cliques of the complete graph $K_n$ on $n$ vertices. We prove that this polytope is $3$-, but not $4$-neighbourly. Following a remark of Pierre Duchet, we partially generalize this result to the $k$-clique polytopes of $r$-uniform complete hypergraphs, $P_{kn}^r$. We show that the neighbourlicity of $P_{kn}^r$ is between $r$ and $2^r-1$ whenever $k\geq r+1$ and $n\geq k+r+1$. Computational results indicate that the upper bound is tight.