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New Proof Settles Decades-Old Bet About Connected Networks

According to mathematical legend, Peter Sarnak and Noga Alon made a bet about optimal graphs in the late 1980s. They’ve now both been proved wrong.

Sarnak proposed that such graphs are rare; he and two collaborators would soon publish a paper that used complicated ideas from number theory to build examples, and he argued that any other constructions would be similarly difficult to achieve. Since mathematicians began studying expander graphs in the 1960s, they’ve been used to model the brain, perform statistical analyses, and build error-correcting codes — encrypted messages that can be read even if they get garbled in transmission. Using a highly technical result in number theory by the Indian math prodigy Srinivasa Ramanujan, Sarnak and his collaborators produced regular graphs that achieved the Alon-Boppana bound.

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