Roth's theorem

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Roth's theorem ({\Bbb Z}/N{\Bbb Z} 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 ({\Bbb Z}/3{\Bbb Z})^n.

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.

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