## This month's topics:

- Ron Graham obituaries
- Dogs' ages in human years: a new rule
- Language learning: an emergent phenomenon?

## Ron Graham obituaries

"Ron Graham Dazzled Admirers With Math and Juggling Feats" was the headline for James Hagerty's July 17, 2020 obituary in the *Wall Street Journal*. Graham, who died July 6, was a pioneer in discrete mathematics. But there was much more, as Hagerty's headline implies. While at the University of Chicago, which he entered at age 15, he "helped support himself by performing in a circus act known as the Bouncing Bears," and he continued kinetic activity his whole life. "Blond, handsome and trim, he was always eager to amaze onlookers and excelled at gymnastics, trampoline tricks, table tennis, flying kites, riding unicycles and throwing boomerangs ... .' Motion stimulated his mind. 'I once had a flash of insight into a stubborn problem in the middle of a back somersault with a triple twist on my trampoline,' he said." Hagerty sketches Graham's mathematical expertise (he mentions Ramsey theory and graph theory) and his friendship and collaboration with Paul Erdös, "a wandering Hungarian mathematical genius;" he leaves us with one of Graham's sobering pronouncements: "The fact that we're humans is really pretty limiting. I mean, we didn't evolve to understand the structure of the space-time continuum or do things in 100,000 dimensions. We know how to stay out of the rain and keep [from] getting eaten by animals."

Kenneth Chang, writing for the *New York Times* (July 23), goes into more mathematical detail. "When they met, Dr. Graham and Dr. Erdös were among the few working in discrete mathematics, particularly in an area known as combinatorics — the mathematics of combinations.

- "In an introductory probability class, a simple combinatorics problem might ask: If one pulls three balls at random out of a bag that contains six blue ones and four red ones, what are the chances that all three are red? (The answer is 1 out of 30.) Combinatorics proved to be important to the rise of digital technology in the 1970s. ...

"[Combinatorics] led to what became known as Graham's number, which was for a time the largest number used in a proof, according to the Guinness Book of World Records. The number came out of a problem [in a field] known as ... Ramsey theory, which states that in large systems there can never be complete disorder, that pockets of structure will appear within the apparent chaos.

- "Dr. Graham was looking at cubes in which the lines between the corners were colored red or blue. In a three-dimensional cube, it is easy to color the lines so that no planar slice of the cube with four vertexes has edges all of one color. But mathematicians can also imagine cubes in four dimensions and greater, and so Dr. Graham wanted to know whether this property of being able to avoid slices of one color would persist in greater dimensions. 'The answer: no,' Dr. Graham explained in 2014 in an episode of Numberphile. 'If the dimension is large enough, you cannot avoid it. No matter how you color it, you cannot avoid it.' No one knows in precisely what dimension this unavoidability would kick in, but Dr. Graham calculated an upper bound for the answer — a number so huge that there is not enough space in the entire universe in which to write all of the digits." [In the video, Graham mentions that dimension 13 might be large enough, but that since the number of red-blue colorings of the edges of a 13-dimensional cube is $2^{33,550,336}$, we may never know.]

## Dogs' ages in human years: a new rule

"Quantitative Translation of Dog-to-Human Aging by Conserved Remodeling of the DNA Methylome," by Trey Ikert (UCSD) and 10 collaborators, was published in *Cell Systems* on July 2, 2020, but the preprint posted on bioRxiv was picked up last November by Virginia Morell for *Science*, with the title: "Here's a better way to convert dog years to human years, scientists say." No more multiplying by 7; the new science-based algorithm is $$\mbox{age}_{\mbox{human}}= 16 \ln(\mbox{age}_{\mbox{dog}}) + 31. $$ (Natural logarithms are more than most people can do in their heads, but TheBark.com —"dog is my copilot"— has a handy table).