An analogue of the Erd\H{o}s-Gallai theorem for random graphs

TL;DR

This paper extends the Erd ext{"o}s-Gallai theorem to random graphs, determining the maximum edges in path-free subgraphs of G(N,p) across various parameters, advancing extremal graph theory in probabilistic settings.

Contribution

It provides a probabilistic analogue of the Erd ext{"o}s-Gallai theorem, quantifying maximum edges in path-free subgraphs within random graphs for broad parameter ranges.

Findings

Determines maximum edges in P_n-free subgraphs of G(N,p) up to a constant factor.

Extends classical extremal results to the random graph setting.

Connects to recent progress on size-Ramsey numbers of paths.

Abstract

Recently, variants of many classical extremal theorems have been proved in the random environment. We, complementing existing results, extend the Erd\H{o}s-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.