Bosio 2026 large b 2 g subsets first

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theorem_1_1: For every fixed integer g >= 1, the first n squares contain a B_2[g] set of size >>_g n^(2g/(2g+1)) (log n)^((2-2^g)/(2g+1)); for g = 1 this is Lefmann and Thiele's n^(2/3) bound for Sidon sets.

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F. Bosio, J. Tarr, R. Riblet, Large B_2[g] subsets of the first squares. arXiv preprint (2026). arXiv:2607.02728. The copy read for this card is arXiv:2607.02728v1 (2 July 2026); the arXiv record also lists a v2 (14 July 2026), not read for this card.

Theorem 1.1 shows that for every fixed integer g >= 1 the largest B_2[g] subset of the first n squares has size at least c_g n^(2g/(2g+1)) (log n)^((2-2^g)/(2g+1)) for all large n, with c_g > 0 depending only on g. The method extends Lefmann and Thiele's hypergraph argument from Sidon sets to general g. The hypergraph on {1, ..., n} has as edges the sets of 2(g+1) integers that split into g+1 pairs with a common sum of squares (4-element edges when g=1), so the squares of an independent set have at most g representations by two distinct elements; the authors bound the numbers of edges and of 2-cycles and apply the Duke-Lefmann-Roedl independence theorem for uncrowded hypergraphs. For g=1 the log power vanishes and the bound recovers Lefmann and Thiele's F(squares) >> n^(2/3), which removed the n^(-epsilon) loss from Alon and Erdos's n^(2/3-epsilon) and does not decide their question whether F(squares) = n^(1-o(1)); for fixed g it also improves the random-deletion bound of Croot, Mao, Pohoata, Sheffer and Yip by replacing a super-polylogarithmic loss with a power of log n (p. 2). For problem 158 this is a finite B_2[g] construction in the squares: rescaled to the ambient endpoint x = n^2 the g=2 bound is x^(2/5)(log x)^(-2/5) up to a constant, and being finite it says nothing about the liminf of A(x)/sqrt(x) for a single infinite set.

Source: https://arxiv.org/abs/2607.02728. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2607.02728), every other right reserved.

Bears on. #773: Theorem 1.1 with g=1 reproves Lefmann and Thiele's lower bound F(squares up to n^2) >> n^(2/3) (p. 2); it is no stronger than that known bound and leaves open whether the answer is n^(1-o(1)). #158: Theorem 1.1 with g=2 gives finite B_2[2] sets in the squares; the paper does not mention the problem, and a finite construction does not address the liminf the problem asks about.

Results. Labels and pages are those of v1.

• Theorem 1.1 (p. 2): B_2[g] subsets of the first n squares of size >>_g n^(2g/(2g+1)) (log n)^((2-2^g)/(2g+1)).

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.