Conlon 2022 rational exponents near two

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Conlon, David and Janzer, Oliver, Rational exponents near two. Adv. Comb. (2022), Paper No. 9, 10 pp. doi:10.19086/aic.2022.9. The arXiv record (https://arxiv.org/abs/2203.03375, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

The Erdos-Simonovits rational exponents conjecture (Conjecture 1.1) asks that every rational r in [1,2] be realizable, meaning ex(n,H) = Theta(n^r) for a single graph H. Theorem 1.2 shows all rationals r = 2 - a/b with b >= max(a, (a-1)^2) are realizable, proving in a strong form a conjecture of Jiang, Jiang and Ma and complementing Jiang-Qiu's result for exponents 1 + p/q with q > p^2. The framework is that of Bukh-Conlon rooted graphs: for a balanced rooted graph F of density rho, the t-blowup satisfies ex(n, F^t) = Omega(n^{2-1/rho}) for all t large enough (Lemma 1.3), and Bukh-Conlon Conjecture 1.4 predicts a matching upper bound for balanced rooted trees. Theorem 1.5 proves this upper bound for the rooted trees F_{r,s} of Jiang, Jiang and Ma when r >= s+2 >= 3 (Jiang, Jiang and Ma needed r >= s^3 - 1); the density (rs+r)/(r+1) of F_{r,s} gives the exponents 2 - (r+1)/(rs+r), and an observation of Kang, Kim and Liu together with earlier cases then yields Theorem 1.2. It is cited for problem 571, which is exactly the rational exponents conjecture, as the state of the art near exponent two.

Source: https://arxiv.org/abs/2203.03375.

Bears on. #571

Results to transcribe.

• Theorem 1.2: All rationals r = 2 - a/b with b >= max(a, (a-1)^2) are realizable as Turan exponents.
• Lemma 1.3 (Bukh-Conlon): For a balanced rooted graph F of density rho there is t0 with ex(n, F^t) = Omega(n^{2-1/rho}) for all t >= t0.
• Conjecture 1.4 (Bukh-Conlon): For every balanced rooted tree F of density rho and all t, ex(n, F^t) = O(n^{2-1/rho}); verified here for the trees F_{r,s} with r >= s+2 >= 3 (Theorem 1.5).
• Theorem 1.5: For all integers r >= s+2 >= 3 and t >= 1, ex(n, F_{r,s}^t) = O(n^{2-(r+1)/(rs+r)}).
• Family F_{r,s}: The rooted tree F_{r,s} (r legs each with s roots) is balanced when s <= r and has density (rs+r)/(r+1), yielding the claimed exponents.