Abstract:
In 1971, Ron Graham posed the following well-known conjecture in
combinatorial group theory, with interesting connections to design theory,
the theory of Latin squares, error-correcting codes, and juggling. Any subset
of ℤ???? \ {0}, with ???? a prime, can be ordered as ????1, ????2, … , ???????? so that all partial
sums ????1 + ????2 + ⋯ + ???????? are distinct. We discuss recent work surrounding this
conjecture, including a proof of the conjecture for dense sets. Combined with
several other recent works, concerned with the sparse case, this gives a full
proof of Graham's Conjecture for any large enough prime ???? . The proof is
based on a synergy between a remarkable number of ideas coming from
several different areas of mathematics, including Fourier-analytic methods
from additive combinatorics, theory of (sublinear) expander graphs, and the
absorption method from extremal combinatorics. Based on a joint work
with: Benjamin Bedert, Noah Kravitz, Richard Montgomery, and Alp
Müyesser.