Triply Existentially Complete Triangle-Free Graphs
Chaim Even-Zohar, Nati Linial

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.
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