Holiday Contest 2026 - Team Round - Problem 10

Call a graph G special if it's vertices can be paritioned into 2025 (possibly empty) sets V_1,V_2,\dots,V_{2025} s.t. each V_i is an independent set, and for any 1 \le i < j < 2025, v \in V_i, w \in V_{j}, v and w are connected iff (i,j) \neq (1,2025). For a complete graph K_n, call a parition of it's edges into sets E_1,E_2,\dots,E_i good if for each E_j, the E_j-induced graph is special. Let f(n) denote the minimum number of sets needed in a good parition of K_n. Find \lim_{n \rightarrow \infty} \dfrac{n}{f(n)}.