---

Claim. Samuel Korsky submitted a partial proof claim on the site's proof-claim tab on 29 July 2026 (the claim's date), naming the AI system GPT 5.6-Pro as a tool (tab accessed 2026-09-19 and 2026-10-06). The claim concerns Problem 934: for every $t\ge3$,

$$h_t(d)\ge(1-o(1))\,d^t\qquad\text{as }d\to\infty,$$

that is, for every $\varepsilon>0$ and every sufficiently large $d$ there is a graph of maximum degree at most $d$ with at least $(1-\varepsilon)d^t$ edges in which every two edges are joined by a path of length less than $t$. This is the lower half of the asymptotic conjecture of Cambie, Cames van Batenburg, de Joannis de Verclos and Kang (Conjecture 3 of their 2022 paper), which asks for the bound along infinitely many $d$ only and which was known for $t\in\{2,3,4,6\}$ from the incidence graphs of generalized polygons. As the summary describes the construction, it takes a prime power $q$ of order $d^{1/(t-1)}$, splits each complete flag of $\mathbb F_q^t$ into the subflag of its odd-rank subspaces and the subflag of its even-rank subspaces, and uses these subflags as the two vertex classes of a regular bipartite graph whose line graph has diameter at most $t$. The tab's summary states the bound as $d\to\infty$ without restricting $d$ to a sequence; the manuscript's range of $d$ (every large $d$ or a sequence) is not recorded on this page.

The same result was posted on 4 August 2026 as the arXiv preprint Asymptotically attaining the Moore bound (arXiv:2608.03965v1, "7+ε pages") by Wouter Cames van Batenburg and Samuel Korsky, announced on the problem's discussion thread on 9 August 2026 and recorded on the problem page as [CvBK26]; its abstract states for every fixed $\ell\ge2$ graphs of maximum degree at most $d$ and line-graph diameter at most $\ell$ with $(1+o(1))d^\ell$ edges, and also the asymptotic determination of the degree--diameter function itself (Bollobás's conjecture), which is not this problem. The two postings are one result under this page; the preprint's first author, Cames van Batenburg, is a coauthor of the 2022 conjecture and wrote on the claim's own thread on 29 July 2026 that they had extended the construction that day, with a preliminary draft they said had survived an audit by ChatGPT 5.6 pro but was not yet prepared to vouch for.

Submission note. Posted to erdosproblems.com as a proof claim by Samuel Korsky (account SamKorsky) on 29 July 2026, giving "GPT 5.6-Pro" as the AI used:

Confirming (and strengthening) a conjecture of Cambie, Cames van Batenburg, de Joannis de Verclos, and Kang, we show that for $t > 2$,

$$h_t(d) \ge (1 - > o(1))d^t \qquad \text{as}\ d\rightarrow\infty$$

using a construction involving

complete flags over $\mathbb F_q^t$ for a prime power $q \sim d^{1/(t - 1)}$, in place of the generalized polygon constructions for $t \in \{2, 3, 4, 6\}$ for which the lower bound was already known. The key is to split a complete flag into its odd- and even-rank subflags and use those subflags as vertices in a cleverly designed regular bipartite graph.

Covers. The lower asymptotic $h_t(d)\ge(1-o(1))d^t$ for every $t\ge3$, and nothing else. With the 2022 upper bound $h_t(d)\le\frac32d^t+1$ this would place $h_t(d)$ between $(1-o(1))d^t$ and $\frac32d^t+1$ for every $t\ge3$, improving the earlier $0.629^td^t$, which held only for large $t$ and infinitely many $d$. It does not estimate $h_t(d)$ up to a factor $1+o(1)$: at $t=3$ the preprint of Kumar, Mohar and Pragada (its claim page) shows $\liminf h_3(d)/d^3\ge\frac{253}{225}$, so the matching upper asymptotic is false there and the leading constant is unknown for every $t\ge3$; and it gives no "nice expression" in Erdős's sense. The site labels the problem OPEN (2026-10-06).

Depends on. Nothing in this wiki: the construction rests on the geometry of flags over finite fields and on the prime powers being dense enough, none of which is recorded in this wiki.

Standing. Claimed: the result is a proof-claim tab submission with a shared-drive file and an arXiv preprint, neither refereed; the tab states that a listing is no guarantee of correctness and does not mean that anyone associated with the site has examined the proof. The claim's thread carries three comments of 29 and 30 July 2026: the coauthor's announcement above, the claimant's reply, and a reader's remark that the construction looks correct and generalizes a lemma of the Kumar--Mohar--Pragada preprint; none is a review, and no named mathematician's acceptance is recorded. No review of the manuscript is recorded.