Bowtie-free graphs have a Ramsey lift

Abstract

A bowtie is a graph consisting of two triangles with one vertex identified. We show that the class of all (finite) graphs not containing a bowtie as a subgraph has a Ramsey lift (expansion). This solves one of the old problems in the area and it is the first Ramsey class with a non-trivial algebraic closure.

Supersaturation Problem for the Bowtie

“…Note that d B (v), d B (w 1 ) ≤ d + 1 and together with (12) we have that d B (w) ≤ d + 2 for any w ∈ V . Therefore, b 1 , b 2 ≤ (1 + o(1))(d + 2)n/4 ≤ dn as d ≥ 1.…”

“…As a consequence, we can derive that the degrees in the graph (V, B) are also close. By (12), this holds in a very strong form for vertices that are in the same part V i . So the following lemma says something new, only if w is not in the same part as u or v. Proof of Theorem 1.…”

“…Note that each bad degree is at most d + 1 + o(d). Indeed, this is true by (12) if V 2 spans no edges. Otherwise, V 2 has two vertices u ′ , v ′ of the same positive degree d ′ and the claim follows in view of a = o(n) from Lemma 19 applied to these two vertices.…”

“…Recall that, by (12), all bad degrees inside a part differ by at most 1. Let d be the maximal integer such that there exist two vertices v, w ∈ V 1 such that d B (v) = d B (w) = d. Since V 1 contains at least one edge, we have d > 0.…”

“…Recall that for every u ∈ V 1 \ {v}, except at most one, we have that d B (u) ≤ d and by (12) for the one exception we have d B (u) ≤ d + 1. When d B (u) ≤ d, we have…”