Universal Graphs with Forbidden Subgraphs and Algebraic Closure

Abstract

We apply model theoretic methods to the problem of existence of countable universal graphs with finitely many forbidden connected subgraphs. We show that to a large extent the question reduces to one of local finiteness of an associated "algebraic closure" operator. The main applications are new examples of universal graphs with forbidden subgraphs and simplified treatments of some previously known cases.

Invariant Measures Concentrated on Countable Structures

“…For example, take C to be the set of cycles of all odd lengths up to a fixed 2n +1. Cherlin et al [15,Theorem 4] have shown that for a homomorphism-closed set C, an existentially complete countable universal C-free graph exists and has trivial algebraic closure. Hence these graphs admit invariant measures.…”

“…For example, K 3 K 3 is the so-called bowtie. An existentially complete countable universal (K m K n )-free graph exists if and only if min(m, n) = 3 or 4, or min(m, n) = 5 but m = n ( [41], [15], and [16]). Any such graph has nontrivial algebraic closure because, by existential completeness, it must contain a copy K of K m+n−2 , but for any vertex v ∈ K, the algebraic closure of {v} in the graph is all of K.…”

“…We apply our results to existing classifications of ultrahomogeneous graphs, directed graphs, and partial orders, as well as other combinatorial structures, thereby providing several new examples of exchangeable constructions that lead to generic structures. Among those structures for which we provide the first such constructions are the countable universal ultrahomogeneous partial order [55] and certain Invariant measures concentrated on countable structures countable universal graphs forbidding a finite homomorphism-closed set of finite connected graphs [15]. Structures for which our results imply the nonexistence of such constructions include Hall's countable universal group [27, Section 7.1, Example 1], and the existentially complete countable universal bowtie-free graph [41].…”

A Combinatorial Classic — Sparse Graphs with High Chromatic Number

“…(The same is true, more generally, for the class of all graphs not containing short odd cycles [18].) In fact one can "forbid" homomorphisms from any finite set of graphs, see [17]. But this does not surprise an interested reader: odd cycles are easier in the whole paper.…”

Universal graphs with a forbidden near‐path or 2‐bouquet

“…We will define this closure operator explicitly in Section 4 in the case of a 2-bouquet. This is based on the general analysis given in [3]. For model theorists this would just be the algebraic closure operator in the class of existentially complete C-free graphs, but in any case, it is necessary to work out explicitly what this means in combinatorial terms in order to make real use of it.…”