Asymptotic enumeration of incidence matrices

We discuss the problem of counting {\em incidence matrices}, i.e. zero-one matrices with no zero rows or columns. Using different approaches we give three different proofs for the leading asymptotics for the number of matrices with $n$ ones as $n\to\infty$. We also give refined results for the asymptotic number of $i\times j$ incidence matrices with $n$ ones.Comment: jpconf style files. Presented at the conference "Counting Complexity: An international workshop on statistical mechanics and combinatorics." In celebration of Prof. Tony Guttmann's 60th birthda

The random graph

Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique (and highly symmetric) countably infinite random graph. This graph, and its automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul Erd\H{o}s

Groups with right-invariant multiorders

A Cayley object for a group G is a structure on which G acts regularly as a group of automorphisms. The main theorem asserts that a necessary and sufficient condition for the free abelian group G of rank m to have the generic n-tuple of linear orders as a Cayley object is that m>n. The background to this theorem is discussed. The proof uses Kronecker's Theorem on diophantine approximation.Comment: 9 page

Two Days of My Six Year War: 9–10 July 1944

Early in July I was called back to Battery H.Q. and briefed on Operation “Charnwood,” after which I went back to the troop and told the sergeants who in turn gave the information to the crews so that everyone knew what was intended and could carry on should he end up as senior man. The gist of the briefing was that an attack was to start by the English on our left at 0700 hours and would proceed south through Buron, Gruchy and Authie just north of Carpiquet airport by mid-morning, after which the Canadian 9th Brigade on the left and the 7th Brigade on the right would move east through their ground and drive to Caen. General FU with whom the division was very familiar and who had a large part in our training in England, was very much in evidence. The English advance was held up and 9th Brigade, according to my notes, moved in and took the villages and 7th Brigade went east to the Abbaye Ardenne and Caen. My recollection of one 48-hour period with the most action of any is as follows