Claims

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1986_03_01_szegedy: Szegedy proves that for every set of n distinct positive integers with n at least an effectively computable n_0 some pair has a/gcd(a,b) at least n, with equality only for the multiples of 1 to n and their reciprocal set.

1987_09_01_zaharescu: Zaharescu proves Graham's conjecture for every chain of n positive integers with n sufficiently large, through the gap below 2n to the largest prime and Huxley's prime-gap bound; refereed in J. Number Theory 27 (1987).

1996_01_01_balasubramanian_soundararajan: Balasubramanian and Soundararajan prove Graham's conjecture for every set of N integers, so every finite set has two members a, b with gcd(a,b) at most a/|A|; refereed in Acta Arithmetica 75 (1996) and credited by the site.

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