On an Extremal Problem for Colored Trees

LetTbe a tree such that there is a propern-coloringcof the vertices ofTwhich, besides a technical condition, isakbkak-free, i.e.,Tcontains no subdivision of a pathu1,…,u3ksuch thatc(u1) = … = c(uk) = c(u2k + 1) = … = c(u3k) ≠ c(uk + 1) = … = c(u2k). ThenThasO(kn) vertices. (The technical condition requires thatTcontains no subdivision of a properly 2-colored starK1,3.) This solves a problem of Klazar, and extends analogous results for generalized Davenport–Schinzel sequences.