Mind Fixers by Anne Harrington

Convex universal fixers

In [1] Burger and Mynhardt introduced the idea of universal fixers. Let G = (V,E) be a graph with n vertices and G a copy of G. For a bijective function π : V (G) → V (G), define the prism πG of G as follows: V (πG) = V (G)∪ V (G) and E(πG) = E(G)∪E(G′)∪Mπ, where Mπ = {uπ(u) | u ∈ V (G)}. Let γ(G) be the domination number of G. If γ(πG) = γ(G) for any bijective function π, then G is called a un...

Characterizing Cartesian fixers and multipliers

Let G H denote the Cartesian product of the graphs G and H. In 2004, Hartnell and Rall [On dominating the Cartesian product of a graph and K2, Discuss. Math. Graph Theory 24(3) (2004), 389–402] characterized prism fixers, i.e., graphs G for which γ(G K2) = γ(G), and noted that γ(G Kn) ≥ min{|V (G)|, γ(G)+n−2}. We call a graph G a consistent fixer if γ(G Kn) = γ(G)+n−2 for each n such that 2 ≤ n...

To Anne

The Paris-Harrington Theorem

In Ramsey theory, very large numbers and fast-growing functions are more of a rule than an exception. The classical Ramsey numbers R(n,m) are known to be of exponential size: the original proof directly gives the upper bound R(n,m) ≤ ( m+n−2 n−1 ) , and an exponential lower bound is also known. For the van der Waerden numbers, the original proof produced upper bounds that were not even primitiv...

2 Leo Harrington And

1. Introduction A set A of nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements.