Infinite paths and cliques in random graphs

TL;DR

This paper investigates conditions for the existence of finite or infinite paths and cliques in random subgraphs of an infinite complete graph, using advanced probabilistic and topological methods without assuming independence.

Contribution

It introduces sharp criteria for percolation phenomena in infinite graphs using topological Ramsey theory and ergodic theory, without relying on independence assumptions.

Findings

Established conditions for the emergence of infinite paths and cliques

Developed a topological Ramsey theory approach for random graphs

Provided new insights into percolation in non-independent settings

Abstract

We study some percolation problems on the complete graph over $\mathbf N$. In particular, we give sharp sufficient conditions for the existence of (finite or infinite) cliques and paths in a random subgraph. No specific assumption on the probability, such as independency, is made. The main tools are a topological version of Ramsey theory, exchangeability theory and elementary ergodic theory.