# Roth's theorem

### From Polymath1Wiki

**Roth's theorem** ( version) If N is sufficiently large depending on δ > 0, then any subset A of [*N*] of density at least δ contains an arithmetic progression x, x+r, x+2r with *r* > 0.

**Roth-Meshulam theorem** ([3]^{n} version) If n is sufficiently large depending on δ > 0, then any subset of [3]^{n} of density at least δ contains an algebraic line, i.e. a triple (x,x+r,x+2r) where r is non-zero and we identify [3]^{n} with .

Roth's theorem is a special case of Szemerédi's theorem. See also Szemerédi's combinatorial proof of Roth's theorem.

Roth's theorem is implied by the corners theorem, which in turn is implied by the k=3 case of the IP-Szemerédi theorem, which is in turn implied by k=3 case of the density Hales-Jewett theorem.