On Rainbow Turán Densities of Trees

ABSTRACT For a given collection of graphs on a common vertex set , which we call a graph system , a graph on a vertex set is called a rainbow subgraph of if there exists an injective function such that for each . The maximum value of over ‐vertex graph systems having no rainbow subgraph isomorphic to is called the rainbow Turán number of . In this article, we study the rainbow Turán density of a tree . While the classical Turán density of a graph lies in the set , the rainbow Turán density exhibits different behaviors because it can even be an irrational number. Nevertheless, we conjecture that the rainbow Turán density is always an algebraic number. We provide evidence for this conjecture by proving that the rainbow Turán density of a tree is an algebraic number. To show this, we identify the structure of extremal graphs for rainbow trees. Moreover, we further determine all tuples such that every graph system satisfying contains all rainbow ‐edge trees. In the course of proving these results, we also develop the theory on the limit of graph systems.