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

Abstract

Abstract:We consider the problem of the existence of universal countable C-free graphs with C a connected finite graph. For C a tree arising by from a path by adjunction of one additional edge we show that a universal countable C-free graph exists. We determine precisely the 2-bouquets C (i.e., unions of two complete graphs with jost one point in common) for which a universal countable C-free graph exists. We lay out some elements of a program for determining all the connected finite graphs C for which a countable universal C-free graph exists. One element of this program is the Tree Conjecture, which is now proved [2]. Our methods involve a mixture of model theory and combinatorics, with the -system lemma playing a significant role.

All those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms)

“…Here we use it to obtain Ramsey lifts of classes defined by forbidden monomorphisms in a greater generality. This extends the family of known Ramsey classes by non-trivial new examples, such as forbidden 2-bouquets [CT07], paths [KMP88,CSS99], complete graphs adjacent to a path [KMP88,CSS99], bowties adjacent to a path [CSS99] and in fact all known cases in the work-in-progress catalogue [CS]. Some of these classes are really exotic ones.…”

Invariant Measures Concentrated on Countable Structures

“…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.…”

Universal graphs with a forbidden subtree

“…The implication (3) ⇒ (1) requires argument, and is treated in [4]. This has a completely different character from anything we do here, lying on the positive side; all of our work here fills in the gap on the negative side.…”

“…Komjáth showed, unexpectedly, that the 2-bouquet formed by joining two triangles over a common vertex provides another constraint graph B allowing a (strongly) universal B-free graph [9]; more generally, 2-bouquets B m,n formed by joining complete graphs K m and K n over a single common vertex have been thoroughly analyzed in [4]: there is a (weakly or strongly) universal B m,n -free graph if and only if the parameters satisfy the following conditions: min(m, n) 5 (m, n) = (5,5). This is visibly a delicate condition, and requires a close combinatorial analysis to achieve.…”