Method of indicators in model theory reference request

Question

I was reading the Wikipedia page on requested articles for mathematical logic. https://en.wikipedia.org/wiki/Wikipedia:Requested_articles/Mathematics/Logic

One requested article is the method of indicators, with the description:

Model-theoretic method used to prove the Paris–Harrington theorem without diagonalization

I could not find a good reference that explains this method. Could someone give me some references?

I read this question and looked up Boolean Relation Theory by Harvey Friedman, but I could not find information about this proof method.

Maybe the method is also known under another similar name, like indicator theory, or just indicators.

Answer 1

I wasn't familiar with this terminology either. But a bit of googling led me to this answer, where Andrés Caicedo gives a reference to the book Metamathematics of First-Order Arithmetic by Petr Hájek and Pavel Pudlák. Section IV.3(b) on p.247 is called Indicators and Paris Sequences and seems to give an explanation of the method.

Maybe someone who is more knowledgeable than me can give a summary of the main ideas.