We examine the minimal hypergraphs which are not 2-colourable, which we call 'condensers'. We show that no condenser has fewer edges than vertices, and we find an odd-cycle characterisation of the condensers which attain equality. We also find explicitly all those condensers attaining equality having the property that any pair of edges intersect.

This paper is a revision of part of my M.Sc. dissertation, prepared under the supervision of Dr A W Ingleton.

Anthony J W Hilton conjectured that if P, Q are collections of subsets of a finite set S, with $|S| = n$, and $|P| >2^{n-2}, |Q| ≥ 2^{n-2}$, then for some $A \in P, B \in Q$ we have $A \subseteq B$ or $B \subseteq A$. We here show that this assertion, indeed a stronger one, can be deduced from a result of D J Kleitman. We then give another proof of a recent result also proved by Lovász and by Schonheim.

The max-flow min-cut theorem of Ford and Fulkerson (for undirected networks) may be regarded as a statement about the circuits and cocircuits using some fixed element of the cycle matroid of a graph. We show that, in general, a matroid has this property (in the integer form) if and only if it is binary and has no minor isomorphic to the dual of the Fano matroid.

Mr and Mrs Richard M MacDonald announce the marriage of their daughter, Shelley Jane, to Dr Paul Douglas Seymour, son of Mr and Mrs Percy Seymour of Plymouth, Devon, England. The wedding took place on 25 August 1979 at Knox Presbyterian Church, Ottawa. The bride's sister, Lelia MacDonald was bridesmaid and Leonard Seymour was best man for his brother. Dr and Mrs Seymour will reside in Oxford, England, during the coming year.

In 1980 he moved to Ohio State University, and began work with Neil Robertson. This led eventually to Seymour's most important accomplishment, the so-called "Graph Minors Project", a series of 23 papers (joint with Robertson), published over the next thirty years, with several significant results: the graph minors structure theorem, that for any fixed graph, all graphs that do not contain it as a minor can be built from graphs that are essentially of bounded genus by piecing them together at small cutsets in a tree structure; a proof of a conjecture of Wagner that in any infinite set of graphs, one of them is a minor of another (and consequently that any property of graphs that can be characterised by excluded minors can be characterised by a finite list of excluded minors); a proof of a similar conjecture of Nash-Williams that in any infinite set of graphs, one of them can be immersed in another; and polynomial-time algorithms to test if a graph contains a fixed graph as a minor, and to solve the k vertex-disjoint paths problem for all fixed k.

Robin Thomas, a renowned mathematician, passed away on 26 March 2020, following a long struggle against Amyotrophic Lateral Sclerosis (ALS). He was born in Prague, Czechoslovakia on 22 August 1962 and earned his doctoral degree in 1985 from Charles University, Prague. Following an invitation from Neil Robertson and Paul Seymour, Robin arrived in the United States in 1988 and had positions at Ohio State University and Bellcore. He joined Georgia Tech (Georgia Institute of Technology, Atlanta, Georgia) in 1989 as a faculty member and was appointed a Regent's Professor in 2010. In 2016, he received the Class of 1934 Distinguished Professor Award, the highest honour for a professor at Georgia Tech.

In about 1990 Robin Thomas began to work with Robertson and Seymour. Their collaboration resulted in several important joint papers over the next ten years: a proof of a conjecture of Sachs, characterising by excluded minors the graphs that admit linkless embeddings in 3-space; a proof that every graph that is not five-colourable has a six-vertex complete graph as a minor (the four-colour theorem is assumed to obtain this result, which is a case of Hadwiger's conjecture); with Dan Sanders, a new, simplified, computer based proof of the four-colour theorem; a description of the bipartite graphs that admit Pfaffian orientations; and the reduction to the "almost-planar" case of a conjecture of Tutte that every bridgeless cubic graph that is not three-edge-colourable contains the Petersen graph as a minor. (The remaining "almost-planar" case has now been solved, completing the proof of Tutte's conjecture, in papers by subsets of Sanders, Seymour, Thomas and Katherine Edwards. This does not assume the four-colour theorem, and re-proves it in an extended form).

A graph is perfect if for every induced subgraph, the chromatic number is equal to the maximum size of a complete subgraph. The class of perfect graphs is important for several reasons. For instance, many problems of interest in practice but intractable in general can be solved efficiently when restricted to the class of perfect graphs. Also, the question of when a certain class of linear programs always have an integer solution can be answered in terms of perfection of an associated graph.

In the first part of the paper we survey the main aspects of perfect graphs and their relevance. In the second part we outline our recent proof of the Strong Perfect Graph Conjecture of Berge from 1961, the following: a graph is perfect if and only if it has no induced subgraph isomorphic to an odd cycle of length at least five, or the complement of such an odd cycle.