Limits RegistryPUBLIC RECORD
Limits RegistryIn May 2016, Ernie Croot, Vsevolod Lev, and Péter Pál Pach introduced a new polynomial method that, combined independently within days by Jordan Ellenberg and Dion Gijswijt, proved a sharp exponential upper bound on cap set size — resolving a growth-rate question about the cap set problem that had resisted attack via traditional Fourier-analytic methods for years. The result also resolved related problems, including bounding the size of 'sunflower-free' families and progression-free sets in other groups.
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In May 2016, Ernie Croot, Vsevolod Lev, and Péter Pál Pach introduced a new polynomial method that, combined independently within days by Jordan Ellenberg and Dion Gijswijt, proved a sharp exponential upper bound on cap set size — resolving a growth-rate question about the cap set problem that had resisted attack via traditional Fourier-analytic methods for years. The result also resolved related problems, including bounding the size of 'sunflower-free' families and progression-free sets in other groups.
The maximum size of a subset of F_3^n containing no three-term arithmetic progression (a 'cap set') grows as O(c^n) for some constant c strictly less than 3, as n tends to infinity.
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Permanent ID limitsregistry.com/limits/LR-CAP-SET-PROBLEM
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Limits Registry. LR-CAP-SET-PROBLEM. Resolution of the Cap Set Problem growth rate. 2026.