Universal graphs without large cliques

TL;DR

This paper explores the existence of universal graphs without large cliques under GCH, providing necessary and sufficient conditions, and investigates how many such graphs embed all others when they do not exist.

Contribution

It establishes new existence and nonexistence criteria for universal K(kappa)-free graphs and analyzes the number of graphs embedding all others under various set-theoretic assumptions.

Findings

Under GCH, characterizes when universal graphs without large cliques exist.

Shows that if no universal K(kappa)-free graph exists, the number of graphs embedding all others can be either minimal or maximal.

Demonstrates consistency results where the number of such graphs varies significantly depending on set-theoretic assumptions.

Abstract

We give some existence/nonexistence statements on universal graphs, which under GCH give a necessary and sufficient condition for the existence of a universal graph of size lambda with no K(kappa), namely, if either kappa is finite or cf(kappa)>cf(lambda). (Here K(kappa) denotes the complete graph on kappa vertices.) The special case when lambda^{< kappa}= lambda was first proved by F. Galvin. Next, we investigate the question that if there is no universal K(kappa)-free graph of size lambda then how many of these graphs embed all the other. It was known, that if lambda^{< lambda}= lambda (e.g., if lambda is regular and the GCH holds below lambda), and kappa = omega, then this number is lambda^+. We show that this holds for every kappa <= lambda of countable cofinality. On the other hand, even for kappa = omega_1, and any regular lambda >= omega_1 it is consistent that the GCH holds below lambda, 2^{lambda} is as large as we wish, and the above number is either lambda^+ or 2^{lambda}, so both extremes can actually occur.