Let $A_1, A_2, \dots, A_n$ be $n$ points on the plane, no three collinear. Each of the segments connecting two points are colored by one of four given colors.
Find the largest natural number $n$ satisfying the following condition: there exists a way to color these segments such that for any four of $n$ points, the segments connecting them (6 segments) are colored by all four colors.
My attempt: I named 4 colors $M_1, M_2, M_3, M_4$, and denote the color of $A_iA_j$ is $M_{ij}$ and consider the number the groups $(A_i, A_j, A_k, A_l, M_{ij}, M_{jk}, M_{kl}, M_{li}, M_{ik}, M_{jl})$. There are $(^n_4)$ ways to choose 4 out of n points, and $4! (^4_1) + 2^2 (^4_2) = 120$ ways to choose the groups of $M_{ij}$ etc, so the number of these groups is $S = 120(^n_4)$ But yet I haven't found the boundary for this number.