Claims

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1966_01_01_folkman: Folkman (Canad. J. Math. 1966) proves that a strictly increasing sequence with a_n < M n^{1+a} for some 0 < a < 1 is subcomplete, which answers the question yes for sets with ≫ N^{1/2+ε} elements up to N; refereed.

2003_01_01_chen: Y.-G. Chen (Acta Arith. 2003) proves that for an absolute constant C every increasing sequence with more than C√n terms up to n, for all n > C², is subcomplete; refereed, not credited by the site.

2005_07_26_szemeredi_vu: Szemerédi and Vu (Ann. of Math. 2006; J. Amer. Math. Soc. 2006) prove that for an absolute constant c every increasing sequence with at least c√n terms up to n is subcomplete; refereed, site-accepted.

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