We consider the random graph M n ¯ p on the set [n], were the probability of {x, y} being an edge is p |x−y| , and ¯ p = (p 1 , p 2 , p 3 , ...) is a series of probabilities. We consider the set of all ¯ q derived from ¯ p by inserting 0 probabilities to ¯ p, or alternatively by decreasing some of the p i. We say that ¯ p hereditarily satisfies the 0-1 law if the 0-1 law (for first order logic)...

We obtain several results and examples concerning the general question “When must a space with a small diagonal have a Gδ-diagonal?”. In particular, we show (1) every compact metrizably fibered space with a small diagonal is metrizable; (2) there are consistent examples of regular Lindelöf (even hereditarily Lindelöf) spaces with a small diagonal but no Gδ-diagonal; (3) every first-countable he...