Balancing a multipartite edge-triangle objective ($F={\sum }_{i<j}{a}_i{a}_j-c{\sum }_{i<j<k}{a}_i{a}_j{a}_k$)

With two part sizes of fixed sum and all other sizes fixed, this objective has form constant plus $(1-cB)a_i{a}_j$, where $B$ is the sum of the other sizes. A nonpositive coefficient permits merging the pair, while a positive coefficient favours balancing it. Choose an optimum with fewest nonzero parts: all remaining coefficients are positive, and the integer sizes differ by at most one. For real sizes they are exactly equal. This explains extremizers among Turan graphs.