An analogue of the Erdos-Gallai theorem for random graphs

Recently, variants of many classical extremal theorems have been proved in the random environment. We, complementing existing results, extend the Erdos-Gallai Theorem in random graphs. In particular, we determine, up to a constant factor, the maximum number of edges in a P-n-free subgraph of G(N, p), practically for all values of N, n and p. Our work is also motivated by the recent progress on the size-Ramsey number of paths. (C) 2020 Elsevier Ltd. All rights reserved.