Graphs without a rainbow path of length 3

In 1959 Erd\H{o}s and Gallai proved the asymptotically optimal bound for the maximum number of edges in graphs not containing a path of a fixed length. Here we study a rainbow version of their theorem, in which one considers $k \geq 1$ graphs on a common set of vertices not creating a path having edges from different graphs and asks for the maximal number of edges in each graph. We prove the asymptotically optimal bound in the case of a path on three edges and any $k \geq 1$

Rainbow Turán Problems for Paths and Forests of Stars

For a fixed graph F, we would like to determine the maximum number of edges in a properly edge-colored graph on n vertices which does not contain a rainbow copy of F, that is, a copy of F all of whose edges receive a different color. This maximum, denoted by ex∗ (n, F), is the rainbow Turán number of F, and its systematic study was initiated by Keevash, Mubayi, Sudakov and Verstraëte [Combinatorics, Probability and Computing 16 (2007)]. We determine ex∗ (n, F) exactly when F is a forest of stars, and give bounds on ex∗ (n, F) when F is a path with l edges, disproving a conjecture in the aforementioned paper for l = 4