Greedy clique decompositions and the Turلn numbers

In this paper, we show that, for p ⩾ 4, if E is a clique decomposition obtained by removing maximal cliques (i.e. their edges) of order at least p one by one until none remain, in which case the remaining edges are removed one by one, then the sum of the orders of the cliques in E is at most 2tp−1(n). Here tp−1(n) denotes the number of edges in the Turán graph of order n which has no cliques of order p.