Consider an infinite graph. In an article by Jonasson and Steif, the Ising model on this graph is defined as a generalization of the standard Ising model. On page 551, they cite a book by Georgii for the fact that there exists at least one Gibbs state. The book is called 'Gibbs measures and phase transitions' and can be downloaded by searching this title on Google followed by 'amu'. I checked page 71 where Jonasson and Steif referred to. However, at this page I found a theorem which contains a lot of unfamiliar terms.
The theorem does not contain the word graph, so I could not conclude to family of graphs the theorem applies.
Secondly, the word Ising model is not mentioned in the theorem and also it is not mentioned in chapter 4 which the theorem is in. The previous chapter contains a section about the one-dimensional Ising model; however, this is not the same as the Ising model on a general infinite graph.
Does the theorem on page 71 indeed state that the Ising model on an infinite graph has a Gibbs state? Does this really work for all graphs, also when they are not connected and/or uncountable?
Also, I am not sure whether this question belongs here or on Mathoverflow. Please let me know what you think.