Ma 2023 upper bounds extremal number 4 cycle

../

corollary_1_4: The paper prints an asymptotic formula for the quadrilateral-free extremal number at q^2+q+1-r for prime powers q, whose displayed proof gives a weaker two-sided bracket.

theorem_1_2: A positive-density set of orders has quadrilateral-free extremal number at most n^{3/2}/2+(1/4-epsilon)n for some fixed positive epsilon.

theorem_1_3: For sufficiently large r at most 0.01q, the quadrilateral-free extremal number at q^2+q+1-r is at most q(q+1)^2/2-0.92rq, for every integer q.

theorem_1_5: For sufficiently large n=q^2+q+1+r with r at most 0.6q, the quadrilateral-free extremal number is at most (q^2+q+1+max{r,2r-0.3q})(q+1)/2, for every integer q.

---

Jie Ma and Tianchi Yang, Upper bounds on the extremal number of the 4-cycle, Bull. Lond. Math. Soc. 55(4) (2023), 1655-1667, DOI 10.1112/blms.12810.

Source and version. The copy read for this card identifies itself as arXiv:2107.11601v3, dated 12 October 2021. It has eleven manuscript pages, with printed and PDF pagination agreeing; these are not the journal's page numbers. On 9 September 2026, the arXiv record still identified v3 as its latest revision. The publisher's metadata confirms publication on 17 February 2023 and the abstract's second-order disproof; the publisher's full proof was not compared with the arXiv manuscript. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2107.11601), every other right reserved.

For finite simple graphs with no $C_4$ as a subgraph, the paper recalls the Kővári-Sós-Turán/Reiman upper bound $\operatorname{ex}(n,C_4)\leq n(1+\sqrt{4n-3})/4$ on p. 1. It also recalls the polarity-graph construction and the leading asymptotic $\operatorname{ex}(n,C_4)\sim n^{3/2}/2$. Equation (3) there reports Füredi's upper bound $q(q+1)^2/2$ at order $q^2+q+1$ for every integer $q\geq14$, citing the 1983 and 1996 papers. Combined with the polarity construction, this gives equality at those orders when $q$ is a prime power. The 1996 proof has not been inspected here; the 1983 Theorem proves the power-of-two case directly.

The paper's new Theorem 1.2 states that, for some fixed $\varepsilon>0$ and a positive-density set of integers $n$,

$$\operatorname{ex}(n,C_4) \leq\frac12n^{3/2}+\left(\frac14-\varepsilon\right)n.$$

The paragraph after the theorem reports any $0<\varepsilon<0.075$ as available. This disproves the proposed formula $n^{3/2}/2+n/4+o(n)$, stated as Conjecture 1.1, and also excludes an $O(n^{1/2})$ remainder. It does not alter the leading asymptotic. The source's proof of Theorem 1.2 on p. 3 uses its nearby-order upper bounds, not Corollary 1.4.

Nearby-order bounds. For integer $q\geq0$, the source defines $I_q^- =\{q^2+1,\ldots,q^2+q\}$ and $I_q^+ =\{q^2+q+2,\ldots,(q+1)^2\}$. Theorems 1.3 and 1.5 on p. 2 do not require $q$ to be a prime power:

• Theorem 1.3 takes $n=q^2+q+1-r\in I_q^-$, with $r\leq0.01q$ sufficiently large, and gives $\operatorname{ex}(n,C_4)\leq q(q+1)^2/2-0.92rq$.
• Theorem 1.5 takes $n=q^2+q+1+r\in I_q^+$ sufficiently large, with $r\leq0.6q$, and gives $\operatorname{ex}(n,C_4)\leq (q^2+q+1+\max{r,2r-0.3q})(q+1)/2$.

Their proofs use counting and structural arguments about degrees and common neighbors, with polynomial inequalities in the appendices. These proofs have not been fully reconstructed or independently checked here.

Corollary 1.4: printed statement and proof-scope gap. On p. 2, v3 prints, in Corollary 1.4, for a prime power $q$ and sufficiently large $r=o(q)$,

$$\operatorname{ex}(q^2+q+1-r,C_4) =\frac12q(q+1)^2-(r+o(1))q.$$

The displayed proof on p. 8, however, concludes only the bracket

$$\frac12q(q+1)^2-rq \leq\operatorname{ex}(q^2+q+1-r,C_4) \leq\frac12q(q+1)^2-(1-\varepsilon)rq,$$

under $r=O(\varepsilon q)$ and $r=\Omega(1/\varepsilon)$, using the claimed refinement (13). This controls the deficit relative to $rq$; it does not by itself give the printed additive $o(q)$ error when $r$ grows. That apparent mismatch remains unresolved here. It is neither a proved correction to the corollary nor a refutation of its statement. The stronger printed formula is not used in the E0765 account, and this gap does not affect the separate application of Theorem 1.2.

Reading and proof scope. All eleven manuscript pages were read. The statements of Theorems 1.2, 1.3 and 1.5 and Corollary 1.4 were checked clause by clause, together with Section 2's disproof route, Corollary 1.4's proof bracket, concluding remarks and references; the proofs in Sections 3-5 were read for structure only. The structural lemmas, the full proofs of Theorems 1.3 and 1.5, and appendix inequalities were not reconstructed or independently reviewed. This is statement-level source compilation with the displayed gap retained, not independent full-proof acceptance or native formalization.

Bears on. #765, by refuting the proposed linear second term through Theorem 1.2, which rests on Theorems 1.3 and 1.5; the catalog's leading-asymptotic request is already supplied by Erdős-Rényi-Sós, Corollary 2.

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