Highest rank of a polytope for $A_n$
  We prove that the highest rank of a string C-group constructed from an alternating group $Alt_n$ is 0 if $n=3, 4, 6, 7, 8$; 3 if $n=5$; 4 if $n=9$; 5 if $n=10$; 6 if $n=11$; and $\lfloor\frac{n-1}{2}\rfloor$ if $n\geq 12$. This solves a conjecture made by the last three authors in 2012. 
Peter J. Cameron, Maria Elisa Fernandes, Dimitri Leemans and Mark   Mixer
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