The absence of monochromatic triangle implies various properly colored spanning trees

An edge-colored graph G is called properly colored if every two adjacent edges are assigned different colors. A monochromatic triangle is a cycle of length 3 with all the edges having the same color. Given a tree T₀, let T(n,T₀) be the collection of n-vertex trees that are subdivisions of T₀. It is conjectured that for each fixed tree T₀ of k edges, there is a function $f(k)$ such that for each integer n≥ f(k) and each T∈ T(n,T₀), every edge-colored complete graph Kₙ without containing monochromatic triangle must contain a properly colored copy of T. We confirm the conjecture in the case that T₀ is a star. A weaker version of the above conjecture is also obtained. Moreover, to get a nice quantitative estimation of $f(k)$ requires determining the constraint Ramsey number of a monochromatic triangle and a rainbow k-star, which is of independent interest.