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https://huji.zoom.us/j/83664008003?pwd=eUdVK2xSUWU4ckFwdVp3SGErNWN3dz09

 
Title: First order of random graph with edge probability inverse of a-power of distance.
Abstract:

For every n let K_n be the set of all finite graphs with size n. 
Given a sequence of probability distributions mu_n : K_n -> [0,1] we call G = ((K_n,mu_n) :n < omega) a 0-1 context.
If M_n is a random graph for the 0-1 context G then, we say that the 0-1 law holds for G if the sequence 
(Prob(M_n models phi) : n < omega) is converge to 0 or 1 where Prob(M_n models phi) = Sum(mu(M): M in K_n and M models phi).
In this joint work with S. Shelah (and based on his previous work in the area) we are proving the 0-1 law for the following 0-1 context in which M_n is  the random graph on [n] = {1,...,n} with the possible edge {i,j} having probability being p_|i-j| = 1/|i-j|^alpha for alpha in (0,1) irrational.   
We will present an abstract frame for analyzing this problem and sufficient conditions for a general 0-1 context to satisfy the 0-1 law. We do it by adding a closure operation, cl, to the general 0-1 context and then show that, under certain conditions, we can guarantee that the for every first order formula phi(x), there is another formula psi_phi(x), such that for any model M and any tuple a in M,  the satisfaction of phi(x) is equivalent to the satisfaction of psi_phi(x) inside the closure of a, i.e. inside cl(a). 
Then we will show how to implement this method to our context. 

If time permits we will talk about generalizations for this problem in various ways.