Triply Existentially Complete Triangle-Free Graphs

TL;DR

This paper explores the existence and construction of 3-existentially complete triangle-free graphs, addressing Cherlin's question about their existence for all k, and provides new and known examples.

Contribution

It introduces new constructions of 3-existentially complete triangle-free graphs and discusses their properties in relation to Cherlin's question.

Findings

Constructed new examples of 3-existentially complete triangle-free graphs.

Reviewed known constructions and their limitations.

Addressed the existence question for k=3 in the context of Cherlin's problem.

Abstract

A triangle-free graph G is called k-existentially complete if for every induced k-vertex subgraph H of G, every extension of H to a (k+1)-vertex triangle-free graph can be realized by adding another vertex of G to H. Cherlin asked whether k-existentially complete triangle-free graphs exist for every k. Here we present known and new constructions of 3-existentially complete triangle-free graphs.