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Claim. J. L. Leonard, On a conjecture of Bollobás and Erdős, Period. Math. Hungar. 3 (1973), no. 3--4, 281--284, exhibits a graph $G$ with $57$ points and $141$ edges in which no two points are joined by five internally disjoint paths (pp. 281--282). Since $57=1+14\cdot4$ and $141=1+14\binom52$, this is the question of Problem 915 at $m=5$, $n=14$ under the vertex-disjoint reading, and the answer there is no. The paper goes on (pp. 282--283) to build, for every integer $s$, graphs with $n$ points and more than $\lfloor5n/2\rfloor+s$ edges and no such pair, so the threshold $k_5(n)$ is not $\frac52n$ plus a constant. Leonard writes that he suspects the edge-disjoint form of the conjecture to be true, and his earlier paper of 1972 had proved it at $m=5$.

The page targets the vertex-disjoint reading of the question, under which the statement is asserted for every $m\ge2$ and $n\ge1$; the counterexample refutes it at $m=5$, $n=14$ and hence as a whole, so the claim is a full disproof. The edge-disjoint reading, under which the conjecture is true for every $m$ by Mader's theorem, is recorded as a variant on Mader's claim page. The disproof for every $m\ge5$, with the exact value of $k_5(n)$, is Sørensen and Thomassen's.

Acceptance. Refereed: Periodica Mathematica Hungarica (volume 3, issue 3--4, pp. 281--284, issued September 1973 by its Crossref record, accessed 2026-10-07; the day is the issue's nominal first day, used for this page's date). Reviewed: the site's curator (T. F. Bloom), independent of the author, credits the paper with the disproof at $m=5$ in the problem's commentary, and Sørensen and Thomassen report the counterexample in the introduction of their 1974 paper (p. 143). The source has a library source card. Read depth: the counterexample and the bound; the constructions are followed, and the clique and path checks are not checked. The acceptance rests on the publication and the site's acceptance; nothing is independently reviewed by this project.