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Claim. The answer to Problem 1006 is no: there is a graph of girth at least five, indeed one of every girth $s\ge5$, with no orientation that has no directed cycle and still has none after any one edge is reversed. The claimed result is Corollary 3 of J. Nešetřil and V. Rödl, On a probabilistic graph-theoretical method, Proc. Amer. Math. Soc. 72 (1978), no. 2, 417--421: for every $s$ there is a graph with no cycle of length less than $s$ which, under every ordering of its vertices, contains a monotone path $v_1v_2\cdots v_s$, $v_1<\dots<v_s$, closed into an $s$-cycle by the edge $v_1v_s$ (the paper's Figure 1). Corollary 4 of the same paper is the Hasse-diagram form: for every $s$, a graph with no cycle shorter than $s$ that is not a subgraph of the Hasse diagram of any partial order. The corpus states both on its result pages Corollary 3 and Corollary 4 (p. 419, with the one-sentence proof of Corollary 4 on p. 420).

How the corollary answers the question. Take $s=5$. The graph has no $C_3$ and no $C_4$. An orientation with no directed cycle is acyclic and has a topological ordering, in which every arc points forward; Corollary 3 applied to that ordering gives a monotone path $v_1\to v_2\to v_3\to v_4\to v_5$ closed by the edge $v_1\to v_5$, and reversing that edge alone closes the directed cycle $v_1\to\dots\to v_5\to v_1$. So no orientation meets both conditions. This deduction is written out on the problem page and is the corpus's own; the paper presents Corollary 3 as the full solution of a problem of Erdős and Ore, citing Erdős's 1971 list, so the deduction is only the step from the paper's ordered-cycle form to the site's orientation form. Every $s\ge5$ gives such a graph of girth $s$, the form in which the site's commentary states the result.

Acceptance. Refereed publication in the Proceedings of the American Mathematical Society (received 20 May 1977, revised 6 January 1978; the Crossref record (2026-10-07) places the paper in volume 72, number 2, issued November 1978, whose nominal first day is this page's date). The site's curator, Thomas Bloom, labels the problem disproved and credits Nešetřil and Rödl [NeRo78b] with the disproof (the reviewed evidence; Bloom took no part in the paper), after the forum comment of 28 September 2025 pointed to Corollary 4 by a link to the paper on the publisher's site; the proof-claim tab is empty. Read depth: claims checked for Corollaries 3 and 4 and Theorem 2; the probabilistic proof of Theorem 2 and its Lemma were read for structure only, and nothing is independently reviewed by this project. The acceptance rests on the publication and the curator's credit.