Stability of the independence number of $G(n, r, 1)$ graphs

TL;DR

This paper establishes a correct proof for the stability of the independence number in $G(n, r, 1)$ graphs, correcting previous inaccuracies and completing the theoretical understanding of this property.

Contribution

The paper provides the first correct proof of the stability theorem for the independence number in $G(n, r, 1)$ graphs, resolving prior errors.

Findings

Proved the stability theorem for $G(n, r, 1)$ graphs.

Corrected the proof of the key lemma from previous work.

Completed the proof of the stability theorem for the independence number.

Abstract

In this paper we obtain the stability theorem for the independence number of $G(n, r, 1)$ graphs. This result was previously stated in the paper of M. Pyaderkin but the proof there was incorrect. We introduce the correct proof of the key lemma and thus finally complete the proof of this theorem.