Jiang 2023 many turan exponents via subdivisions

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Jiang, Tao and Qiu, Yu, Many Turán exponents via subdivisions. Combin. Probab. Comput. 32 (2023), no. 1, 134--150, doi:10.1017/S0963548322000177 (published online 21 July 2022; Crossref record read). The copy read is arXiv:1908.02385v1 (6 August 2019, the only arXiv version), 20 pages; the journal text was not compared, so the labels are the preprint's. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1908.02385), every other right reserved.

The paper establishes a large new family of Turan exponents for single bipartite graphs. Theorem 1.2 shows that 1 + p/(kp+b) is a Turan exponent for all positive integers p, k, b with k >= b, and Theorem 1.3 draws the clean corollary that 1 + p/q is a Turan exponent for all positive integers q > p^2. Corollary 1.4 and Corollary 1.5 transfer these via a reduction lemma of Kang, Kim and Liu to exponents of the form 2 - p/q, in particular for all positive integers q > p with (q mod p) <= sqrt(p). The engine is Theorem 1.10, an upper bound ex(n, t * S_{b,k}^s) = O(n^{1+(s-1)/((s-1)k+b)}) for subdivisions of K_{s,t} in which different edges may be subdivided different numbers of times, which partially answers a conjecture of Janzer; the proof uses the Erdos-Simonovits regularization lemma to pass to almost-regular host graphs together with the recent subdivision-counting machinery. For problem 571, the Erdos-Simonovits rational exponent conjecture, it is a partial result covering all rationals 1 + p/q with q > p^2.

Source: https://arxiv.org/abs/1908.02385.

Bears on. #571: a partial result. Theorem 1.3 (p. 2) realizes every rational 1 + p/q with q > p^2 as the Turan exponent of a single bipartite graph, and Corollary 1.5 (p. 2) every 2 - p/q with q > p and (q mod p) <= sqrt(p); the problem asks for every rational in [1,2), which the paper does not reach.

Results to transcribe.

• Theorem 1.2 (p. 2): 1 + p/(kp+b) is a Turan exponent (ex(n, H) = Theta(n^r) for some bipartite H) for every choice of positive integers p, k, b with k >= b.
• Theorem 1.3 (p. 2): 1 + p/q is a Turan exponent for every pair of positive integers p, q with q > p^2.
• Corollary 1.4 / 1.5 (p. 2): for integers b, p, s >= 1 and k >= 0 with k >= b-1, 2 - (kp+b)/(s(kp+b)+p) is a Turan exponent; hence 2 - p/q for all positive integers q > p with (q mod p) <= sqrt(p).
• Theorem 1.10 (p. 4): For s, t >= 2 and k >= b >= 1, ex(n, t * S_{b,k}^s) = O(n^{1+(s-1)/((s-1)k+b)}), for unevenly subdivided K_{s,t}.

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