On a clique covering problem of orlin

Let T_2n be the complement of a perfect matching in the complete graph on 2n vertices, and cc(T_2n) be the minimum number of complete subgraphs necessary to cover all the edges of T_2n Orlin posed the problem of determining the asymptotic behaviour of cc(T_2n). We show that $\text{cc}\text{(T}{}_{\mathrm{2n}}\text{)=}\text{min}\text{{k:n?(}\text{k?1}\text{?}\text{k}\text{2}\text{?}\text{)}}$ for all n>1, (which implies that lim_n→∞cc(T_2n)/log₂n=1). This is done by applying a Sperner-type theorem on set families due to Bollobás and Schönheim.