# Eric Charles Milner

### Quick Info

London, England

Calgary, Alberta, Canada

**Eric Milner**was an English mathematician who spent much of his time abroad and who worked mainly in combinatorial set theory.

### Biography

**Eric Milner**was the son of Frederick C Milner (1905-1981), born on 28 July 1905 in London, a fitter, turner and engine smith, and Annie Elizabeth Kluth (1905-2000), born 2 June 1905, a mantle machinist. Although at this time it was typical for only the husband to work, nevertheless if he did not bring in enough in wages to support the family the wife would often help out taking in mending or similar work. This is exactly what happened here, for Frederick Milner was often out of work and Annie helped out with the finances as a seamstress. In fact at times his parents were not able to provide the necessary care due to their work and Eric was looked after by his grandmother. He was his parents' only child and he was brought up in Lewisham, in the south part of the city of London where, despite the hardship, he excelled at elementary school. When he was eleven years old he won a scholarship to the Haberdashers' Aske's Boys' School. This independent school had been founded in 1690 and provided a high standard of education. In August 1939 World War II broke out and a Government Evacuation Scheme which had been planned since the summer of 1938 was put into action. He lost his chance of being educated in Haberdashers' Aske's Boys' School.

Eric Milner, like other children evacuated from London, was separated from his family. He was sent to live with a family near Reading where, not only was he separated from his parents, but he had to attend a school where he knew none of the other pupils. Frightened and alone, he ran away and went back to London. The authorities tried to find a more suitable place to send him but this took time and Eric wandered around the streets of London when he should have been at school. Eventually a new place was found for him to live where he was received with kindness and understanding. Despite highly disrupted school years, being a very intelligent boy, he managed to get by and enter King's College, London in 1946 to begin his university studies. By this time World War II had ended and King's College, which had been evacuated to the University of Bristol during the war, was again operating from London. It was a difficult time, however, with some of the buildings having been destroyed in the bombing of London.

At King's College, mathematics was part of the Faculty of Arts and the Faculty of Science when Milner began his studies. George Temple was head of mathematics at King's College, having returned from war work to take up his chair again in 1945. Other members of the mathematics staff at this time were John Semple and George McVittie. Richard Rado joined the department in 1947 and in the same year Charles Coulson became the chair of the Department of Theoretical Physics. Milner graduated with a First Class Honours degree in mathematics in 1949 having received the Drew Gold Medal as top ranked mathematics student in that year. He was awarded a Research Studentship which allowed him to continue his studies at King's College. For his M.Sc. degree he took two topics, Modern algebra and Quantum mechanics. His advisors for these topics were Richard Rado and Charles Coulson. His work was impressive and he was awarded an M.Sc. with distinction in 1950. He then began research on quantum mechanics advised by Coulson, but after a year he left and travelled to Malaya. This may appear to be a very unusual decision by Milner, so to gain some understanding of why he made this decision, we quote from [3]:-

Those who knew Eric only as a mature fellow-mathematician (and perhaps even worked closely with him) may not have realised what a many-sided person he was. A colleague has described him as "a remarkable mixture of Cockney street smartness, wild adventurer and uncompromising mathematician." He had considerable business acumen, and was also very athletic: he was a featherweight boxer for the University of London around 1947, and took immediately to skiing when he was nearly forty. In his early youth, he did not contemplate an academic career: nothing in his family background would have suggested such a possibility. At that time, National Service was compulsory in Britain, and Eric applied to join the Royal Navy, in which his academic record would normally have secured a commission and in which he might well have made his career. He was disappointed to be rejected for the Navy because he was found to be somewhat deaf, a fact which had not previously been noticed. Joining the Army did not appeal to him; but service in other Commonwealth countries was a permitted alternative. Possibly his deafness could have secured exemption from any form of National Service, but the foregoing circumstances and an adventurous spirit may explain a decision to go to Malaya in 1951 to work as a tin assayer for the Straits Trading Company, a tin mining and smelting company.Milner sailed on the ship

*Glenroy*from London to Singapore on his way to Malaya, leaving London on 9 October 1951. He gave his U.K. address as '98 Braidwood Road, Catford, London' and his occupation as 'Commercial Assistant'. At this time Alexander Oppenheim was Dean of the Faculty of Arts of the University of Malaya in Singapore. This university had been founded in October 1949 in Singapore through a merger of King Edward VII College of Medicine and Raffles College. Also in 1951, the mathematician Richard Guy (born 30 September 1916) had joined the Mathematics Department of the University of Malaya. Soon after both Guy and Milner arrived, they met socially and Guy tried hard to persuade Milner to give up his idea of working as a commercial assistant for the Straits Trading Company and join the Mathematics Department of the University of Malaya. Milner took a lot of persuading but eventually decided that he would give it a try.

While he had been a student at King's College, London, Milner had met Esther Stella Lawton (known as Estelle). She had been born on 5 May 1929 to George R Lawton (born 1 March 1900), a Commercial Traveller, and his wife Caroline Marks (born 25 July 1904). Milner married Estelle in the second half of 1954, the marriage being registered in Willesden, Middlesex, England.

Milner had proposed a problem to the

*American Mathematical Monthly*, which was published in February 1953:-

Prove that for any positive integers, $n, N$ there are blocks of consecutive integers of length greater than $N$, with the property that each of their totients is divisible by $n$.The

*totient*of a number $m$ is the number of positive integers less than $m$ which are coprime to $m$.

We see that despite Milner having been doing research on quantum mechanics while in London, he had becoame interested in number theory at the University of Malaya. This, however, is not surprising since both Oppenheim and Guy were interested in number theory and combinatorics. In fact Milner's first research paper was the joint publication

*Generalized decimals*(1955) with Oppenheim. Milner's interest in number theory and combinatorial set theory was greatly increased by Paul Erdős who made several visits to Singapore. In fact it may have been Erdős's influence which finally convinced Milner that a university career in mathematics was right for him. Certainly it was through a suggestion by Erdős that Milner spent his sabbatical leave, 1958-59, at the University of Reading with Richard Rado. Milner knew Rado well from his time at King's College, London, and Rado had moved to Reading to take up the chair of mathematics there in 1954. Another strong influence on Milner's career was Andras Hajnal whom he met in Reading in 1958. The year 1958 was important to Milner in other ways too, since Eric and Estelle's first child Suzanne was born in June of that year. We note at this point that Eric and Estelle Milner had three sons: Mark, Paul and Simon.

Back in Singapore, Milner and Oppenheim published another joint paper,

*Properties of tetrahedra*(1960). He decided, however, that he would be better to try to return to Britain than to remain in Singapore; perhaps partly for family reasons and perhaps partly since there were changes in Singapore as it gained independence from Britain with a new constitution. Richard Rado offered him a lectureship at the University of Reading which he was happy to accept and took up his new post in January 1961. His first two single authored papers were published in 1962, namely

*Chromatic graphs*and

*A theorem of Schur*, both in the

*Bulletin*of the Malaysian Mathematical Society.

Milner had begun a Ph.D. at the University of London before going to Singapore so, back in Britain in a university post, he immediately decided to complete his doctorate, not in quantum mechanics but in his new area of interest, combinatorial set theory. He worked on his thesis as an external University of London student without a formal advisor but, in practice, Richard Rado acted as his advisor. He submitted his thesis

*Some combinatorial problems in set theory*in 1962 and, after an oral examination by Richard Rado and Roy Davies, he was awarded the degree of Ph.D. in 1963. His examiners declared the thesis to be "of an exceptionally high standard."

In October 1964 Milner and Rado submitted a joint paper

*The Pigeon-Hole Principle for Ordinal Numbers*to the London Mathematical Society and it was published in the

*Proceeding*of the Society in 1965. The authors begin the paper as follows:-

Dirichlet's pigeon-hole principle (chest-of-drawers principle, Schubfachprinzip) asserts, roughly, that if a large number of objects is distributed in any way over not too many classes, then one of these classes contains many of these objects. Here we consider an extension of this principle, and investigate distributions of the elements of a well-ordered set into finitely or infinitely many classes.Two of Milner's Singapore colleagues, Richard Guy and Peter Lancaster (born 1929), had moved from Singapore to the University of Calgary in Canada. They tried to persuade Milner to take a position in Calgary and, in 1967, he accepted a professorship at the University of Calgary. Milner spent the rest of his career in Calgary, becoming a Canadian citizen in 1973. He served as head of the Department of Mathematics at Calgary from 1976 to 1980.

When we say that Milner "spent the rest of his career in Calgary" we mean that he continued to hold his permanent professorship there. He loved to travel and he spent time on research visits to other universities around the world: the University of Cambridge, England (1971-72), the University of Tel Aviv, Israel (1972 and 1979 and 1986), Merton College, Oxford, England (1978-79), the University of Singapore, Singapore (1981 and 1984) and the Université Claude Bernard (Lyon I), France (1984 and 1985-86).

Sadly Milner's wife Estelle died of cancer in Calgary on 20 February 1975 at the age of 45. In 1979 Milner married Elizabeth Forsyth Borthwick [3]:-

... a school teacher who had formerly taught Inuit children within the Arctic Circle.Eric and Elizabeth Milner had one son, Robert born in January 1985.

Milner was a plenary 1-hour speaker at the International Congress of Mathematicians held in Vancouver in August 1974. He gave the talk

*Transversal Theory*. He began his Introduction to the lecture as follows:-

Transversal theory is a branch of combinatorial mathematics which is only just beginning to emerge as a reasonably connected and coherent subject. Whether this is yet rich enough or mature enough to be called a 'theory' may be a matter for debate; indeed, it is by no means certain that this part of mathematics may not finally be classified under some broader, more comprehensive title. However, what is beyond dispute is the fact that during the last two decades a large number of papers have been published which include some reference to the so-called marriage theorem, which is the starting point for transversal theory. These papers deal with surprisingly diverse problems and their only connecting link seems to be this common reference to the marriage theorem. The arguments employed have generally had an ad hoc flavour although some of these have been highly original. Transversal theory is a depository for developing those mathematical ideas of the marriage theorem type which frequently recur and which seem to belong to some more general framework.Milner's impressive research contributions are very well presented in [3], where his complete publication list is given, so we will not attempt to give details here. We do note, however, the impressive list of his co-authors including Alexander Oppenheim, Richard Rado, Paul Erdős, Andras Hajnal, Richard Guy, Endre Szemerédi, Béla Bollobás, and Saharon Shelah.

To gain some idea of Milner's approach to mathematics we give some extracts from the talk "The New Math in Paradise", a popular lecture which Milner gave to the Singapore Mathematical Society on 19 June 1991, at THIS LINK.

Among the honours he received we mention his election as a Fellow of the Royal Society of Canada in 1976 and the fact that he was the Canadian Mathematical Society's Jeffery-Williams Lecturer in 1989.

St John Nash-Williams writes about Milner's interests outside mathematics in [3]:-

In addition to demanding professional duties and prolific research, both performed to the highest standards, Eric found time and energy for much else, and enjoyed life to the full. He made two very happy marriages, and his family occupied a great deal of his attention. He was also a devoted son to his mother, who outlived him. People tell with particular warmth of the welcome and hospitality extended to numerous guests in the Milner household (during both of his marriages). His recreations included rugger, tennis, squash, dancing, chess, Go and other board and card games, sailing, mountain walking and ski-trekking. He derived much enjoyment from mountains and, for a number of years, owned an A-frame wooden weekend cottage in Canmore, near Banff, Alberta.Milner's health deteriorated and he was diagnosed with cancer. He retired in 1996 and died in the summer of the following year [3]:-

He will be remembered for outstanding contributions to research and teaching, but particularly, among those closest to him, as a gentle, caring and exceptionally nice man.

### References (show)

- N H Williams,
*Combinatorial set theory*(North-Holland Publishing Company, Amsterdam, 1977). - E C Milner, Transversal theory,
*Proceedings of the International Congress of Mathematicians***1**(Vancouver, BC, 1974), 155-169. - C St J A Nash-Williams, Obituary: Eric Charles Milner,
*Bull. London Math. Soc*.**32**(2000), 91-104.

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Written by J J O'Connor and E F Robertson

Last Update January 2020

Last Update January 2020