Tang 2025 harmonic lcm patterns sunflower free capacity

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corollary_1_7: Tang and Zhang's bounds for k = 3: the largest harmonic sum f_3(N) of a subset of {1,...,N} with no three distinct members of equal pairwise least common multiple satisfies (log N)^(log(1.551) - o(1)) <= f_3(N) << (log N)^(3/2^(2/3) - 1 + o(1)) as N tends to infinity.

theorem_1_2: Tang and Zhang's unconditional lower bound for f_k(N), the largest harmonic sum of a set in {1,...,N} with no k distinct members of equal pairwise least common multiple: for fixed k at least 3, f_k(N) >= (log N)^(c_k - o(1)) as N tends to infinity, where c_k = (k-2)/(e((k-2)!)^(1/(k-2))).

theorem_1_4: Tang and Zhang's equivalence, for each k at least 3: the Erdős-Szemerédi k-sunflower-free capacity equals 2, that is, the Erdős-Szemerédi sunflower conjecture fails at k, if and only if the largest harmonic sum of an LCM-k-free subset of {1,...,N} is (log N)^(1 - o(1)).

theorem_1_5: Tang and Zhang's upper bound for the largest harmonic sum of an LCM-k-free subset of {1,...,N} in terms of the Erdős-Szemerédi k-sunflower-free capacity mu_k^S: for fixed k at least 3, f_k(N) << (log N)^(mu_k^S - 1 + o(1)) as N tends to infinity.

theorem_1_6: Tang and Zhang's lower bound for the largest harmonic sum of an LCM-k-free subset of {1,...,N} in terms of the Erdős-Szemerédi k-sunflower-free capacity mu_k^S: for fixed k at least 3, f_k(N) >= (log N)^(log mu_k^S - o(1)) as N tends to infinity.

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Quanyu Tang, Shengtong Zhang, Harmonic LCM patterns and sunflower-free capacity. arXiv:2512.20055 (2025). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2512.20055), every other right reserved.

Call A in [N] LCM-k-free if no k distinct elements have all pairwise least common multiples equal, and let f_k(N) be the maximal harmonic sum of such a set; estimating f_k(N) is Erdos's question, problem 856. Theorem 1.2 gives the explicit unconditional lower bound f_k(N) >= (log N)^{c_k - o(1)} with c_k = (k-2)/(e ((k-2)!)^{1/(k-2)}), via a prime-bucketing construction that partitions primes into blocks of comparable harmonic sum and takes squarefree integers using exactly k-2 primes from each block; the authors know of no earlier lower bound of comparable strength. Making Erdos's suggested link to the sunflower problem precise, the authors define the Erdos-Szemeredi k-sunflower-free capacity mu_k^S = limsup F_k(n)^{1/n}, with F_k(n) the largest size of a k-sunflower-free family of subsets of [n]; a tensor-power argument shows that this limsup is a limit (their (1.2), p. 3). They prove Theorem 1.5, f_k(N) << (log N)^{mu_k^S - 1 + o(1)}, by a double-counting argument in the spirit of Erdos combined with a Sathe-Selberg lower bound for harmonic sums of squarefree almost primes, and Theorem 1.6, f_k(N) >= (log N)^{log mu_k^S - o(1)}. Theorem 1.4 is the resulting equivalence: for each k >= 3 the Erdos-Szemeredi sunflower conjecture fails at k (that is, mu_k^S = 2) if and only if f_k(N) = (log N)^{1-o(1)}, connecting problem 856 directly to problem 857. Corollary 1.7 makes the k=3 case explicit, (log N)^{log 1.551 - o(1)} <= f_3(N) << (log N)^{3/2^{2/3} - 1 + o(1)}, using the Naslund-Sawin upper bound and the Deuber-Erdos-Gunderson-Kostochka-Meyer construction.

Source: https://arxiv.org/abs/2512.20055.

Read status: claims checked for Theorems 1.2, 1.4, 1.5 and 1.6 and Corollary 1.7, with the definitions they use, read clause by clause on the printed pages of arXiv v1 (23 December 2025); the proofs in Sections 3 to 6 were followed. The results the paper cites (Theorem 5.5 from Alon, Shpilka and Umans, the squarefree Sathe-Selberg theorem behind Lemma 2.2, and the two bounds for mu_3^S) were not checked. Nothing here is independently reviewed.

Bears on.

• #856: the problem's f_k(N) is the paper's. Theorems 1.2, 1.5 and 1.6 bound it between powers of log N, the exponents given by c_k and by mu_k^S; Corollary 1.7 makes the case k = 3 explicit; Theorem 1.4 says f_k(N) = (log N)^{1-o(1)} exactly when mu_k^S = 2. The exponent of f_k(N) is not determined for any k.
• #857: the problem's m(n,k), for distinct sets, is F_k(n) + 1, so mu_k^S is its exponential growth rate. Theorem 1.4 restates whether mu_k^S < 2 as a statement about f_k(N); the paper gives no estimate of m(n,k).

Results.

• Theorem 1.2 (p. 2): for fixed k >= 3, f_k(N) >= (log N)^{c_k - o(1)} with c_k = (k-2)/(e((k-2)!)^{1/(k-2)}), by a prime-bucketing construction.
• Theorem 1.4 (p. 3): for each k >= 3, mu_k^S = 2 if and only if f_k(N) = (log N)^{1-o(1)}.
• Theorem 1.5 (p. 3): for fixed k >= 3, f_k(N) << (log N)^{mu_k^S - 1 + o(1)}.
• Theorem 1.6 (p. 3): for fixed k >= 3, f_k(N) >= (log N)^{log mu_k^S - o(1)}.
• Corollary 1.7 (p. 3): (log N)^{log(1.551) - o(1)} <= f_3(N) << (log N)^{3/2^{2/3} - 1 + o(1)}.
• Prior bounds (p. 2): Erdos proved f_k(N) <<_k log N/log log N; the paper attributes to the problem's site comments the improvement f_k(N) <= log N exp(-Omega_k(log log N/log log log N)), and knows of no earlier lower bound of comparable strength.

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