Elementary combinatorial proof of the complete-graph weak saturation formula

Establish a truly elementary combinatorial proof that, for all integers n≥k, the weak saturation number of the complete graph K_k satisfies w-sat(n,K_k) = {k\choose 2} − 1 + (n−k)(k−2), without relying on matroidal or linear-algebraic methods.

Background

The paper reviews the classical weak saturation problem for complete graphs. Bollobás conjectured the exact formula w-sat(n,K_k) = {k\choose 2} − 1 + (n−k)(k−2), and Lovász subsequently proved it using flats in matroids representable over fields. Several later proofs also used sophisticated algebraic or matroidal techniques, motivating the unresolved question of whether a genuinely elementary combinatorial proof exists.