We consider the random graph $M^n_{\bar{p}}$ on the set $[n]$, were
the probability of $\{x,y\}$ being an edge is $p_{|x-y|}$, and
$\bar{p}=(p_1,p_2,p_3,...)$ is a series of probabilities. We consider
the set of all $\bar{q}$ derived from $\bar{p}$ by inserting 0
probabilities to $\bar{p}$, or alternatively by decreasing some of the
$p_i$. We say that $\bar{p}$ hereditarily satisfies the 0-1 law if the
0-1 law (for first order logic) holds in $M^n_{\bar{q}}$ for any
$\bar{q}$ derived from $\bar{p}$ in the relevant way described
above. We give a necessary and sufficient condition on $\bar{p}$ for
it to hereditarily satisfy the 0-1 law.

The talk is very accessible, with only basic knowledge of Logic and
probability. I will not assume any knowledge of 0-1 laws.