Vertex-minor Ramsey numbers: exact values and extremal structure
Ji Ho Bae

TL;DR
This paper determines the exact vertex-minor Ramsey number for four vertices, classifies extremal graphs avoiding this minor, and explores their structural properties and implications for larger numbers.
Contribution
It provides the first exact value of , classifies extremal graphs, and establishes new lower bounds for s growth, surpassing previous asymptotic bounds.
Findings
=11 is the exact vertex-minor Ramsey number.
Six extremal graphs on 10 vertices avoid as a vertex-minor.
Lower bounds for s are improved, e.g., s lower bound is at least 13.
Abstract
We determine the vertex-minor Ramsey number , where is the smallest~ such that every -vertex graph contains the edgeless graph~ as a vertex-minor. We prove this by an exhaustive classification of the graphs on~ and~ vertices under local complementation. At the extremal order , exactly six non-isomorphic graphs avoid~ as a vertex-minor; up to isomorphism, they represent five LC-equivalence classes, and each labeled LC orbit has cardinality~. Thus is the first case in which the general upper bound is not attained. Using the extremal graphs as building blocks, we derive explicit lower bounds on~ that surpass the leading term of the asymptotic bound for all ; in particular, . We also describe structural properties of the six extremal graphs and formulate the next open problem,…
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