Remembering Uncle Paul
by Fan Chung
(appeared in the Preface of "Erdös on Graphs")

An old friend, Vera S'os, told me perhaps the "last" story about Uncle Paul, an incident which happened at a mathematics conference in Warsaw on September 19, 1996, the day before Paul "left" this world. "Paul told me about some recent joint work with Gy'arfas on multi-colored Ramsey numbers," Vera said. ``I mentioned your related results with Ron in Combinatorica and Paul immediately wanted to telephone Gy'arfas in Budapest." So, Paul had been doing exactly what he liked and wanted to do until the very last day of his life. Throughout the 83 years that he lived, he had been absolutely true to himself beyond any temptation of money and position. Most of us are surrounded by all sorts of worldly comforts and burdens. Every time I saw him it served as a reminder that it is indeed possible to pursue one's dream regardless of all the trivial details in life. For this, I miss Uncle Paul the most.

After the shock of Paul's passing, every combinatorial journal made plans to publish special articles paying tribute to Paul. The editors of the Journal of Graph Theory asked me to write a survey article on open problems of Paul Erdös in graph theory. It was indeed an extremely challenging project since Paul's work has touched upon so many different topics (even when restricted to graph theory only). In preparing the survey, I have exchanged email with hundreds of researchers and colleagues to find the current status and to trace the related references of innumerable problems. In the past, Paul was an excellent source for getting references since he had such an amazing memory. He usually responded instantly to any request. He would say, "Oh, in 1970 or 71, Proceedings of the ...... conference," and he was usually right. How I missed him when I was in need of this huge number of references. His publication list alone involves more than 1500 entries, many of which are nontrivial to find and research through.

When we sorted through Paul's favorite problems, it was often very tempting (and addictive) to start working on some particular problem instead of carrying on with the survey. Ron and I took another look at one of the very earliest of these problems (from 1935), the problem of Erdös and Szekeres on convex n-gons. With a stroke of luck, we were able to improve this sixty-year-old bound. Our first reaction was to think about what Paul would have said if he could have heard this news. He would have been so happy to know about this new development. In fact, his passionate care and relentless pursuit always made it so much more fun to work on and, on occasion, even to solve, his problems. With whom can we share such good news now? We miss you, Paul, for the unique place you will always have in our hearts.

Paul enjoyed playing the game of Go and, in particular, he loved close combat (although sometimes he was very impatient and consequently lost the game). In mathematics, Paul excelled at identifying a sequence of problems; starting from the concrete and essential special cases, which at the same time provide insight to the general problems and push the underlying theory. Working with Paul was like taking a walk in the hills. Every time when I thought that we had achieved our goal and deserved a rest, Paul pointed to the top of another hill and off we would go. His tremendous intuition served as a guiding light and he usually plunged ahead without any hesitation. Sometimes, he could be wrong, but a misstep would only strengthen his determination (and remind us of the humanity of this great man).

Paul's pursuit of mathematics was inseparable from his unique lifestyle of non-stop traveling. He usually had one old briefcase and one beat-up suitcase which contained all his belongings. When Ron helped him pack for the next trip, he sometimes hid in the old suitcase an extra present or some large strange item as a joke between them. Every time Paul arrived at our home, his bags contained the collective activities of the mathematical community. He loved to mention his problems new and old, but he also enjoyed passing along any problem of interest. A substantial part of my work was based on the problems that Paul brought to my attention, in addition to our 13 joint papers and three of his problems that I managed to solve. Paul served as a bridge that joined so many of us together.

My survey paper on the open problems of Paul Erdös in graph theory was written for the Journal of Graph Theory. In this version, references for each problem only included the first source and the last citation of the best known results. As is the nature of such research surveys, there was not enough room to tell the whole story. To do justice to these problems, we have added here more problems and also included some of their history and further development. By continuing to work with these special problems of Erdös, his mathematics and his ``force" will always be with us.


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