This talk will be a review of a paper by Andrew Brooke-Taylor and Damiano Testa:
"The infinite random simplicial complex" (August 27, 2013)
Our aim will be to establish a first order 0-1 law for the class of finite (undirected) simplicial complexes.
We will do so by a known technique involving the Fra¨ıss´e limit of a more general classes called "local classes".
Simplicial complexes are important objects in various fields in mathematics, among them are algebraic topology and combinatorics. One might think about them as "gluing" simplexes along their faces.
0-1 laws are a central subject of research in finite model theory, and many occurrences of those are well studied. We say that a logic L has a 0-1 law with respect to a class of structures K if given a sentence, the probability of randomly choosing a structure from K which satisfy that sentence is always 1 or 0. We will work in order to obtain such a law for the so called local classes which capture the case of simplicial complexes.