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Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

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Claim. For n=10n=10 and n=11n=11, every family of trees T2,…,TnT_2,\ldots,T_n with ∣Tk∣=k|T_k|=k packs into KnK_n. This is the computer verification of David R. Guichard and John D. Massman, A note on packing complete graphs with trees, J. Combin. Math. Combin. Comput. 8 (1990), 123--126 (no DOI; the volume carries a year and no month, so this page is dated to 1990 with a nominal day). The note generates Fishburn's universally recursive families U8\mathcal U_8 and U9\mathcal U_9 by computer and shows directly that the one exceptional tree in each case still packs, "proving the Gyárfás--Lehel conjecture for n=10n=10" and "for n=11n=11" (p. 124); its abstract adds that an approach suggested by Fishburn is unlikely to work in general. The corpus holds the publisher's copy on its card and pages the verification at verification_p124. The note is not a site key; a thread comment of 27 August 2026 named it.

Covers. The instances of Problem 743 with n=10n=10 and n=11n=11, every family of trees included; for them the answer is yes. With Fishburn's verification for n≤9n\le9, every n≤11n\le11 is settled, and n=12n=12 is the first order no published exhaustive check covers.

Depends on. Fishburn's verification: the note follows Fishburn's route, splitting KnK_n into his half-complete graphs Hn−1H_{n-1} and HnH_n (p. 124); for n=10n=10 the odd-order trees T3,…,T9T_3,\ldots,T_9 pack into H9H_9 by Fishburn's H9∈U9H_9\in\mathcal U_9, and the families U8\mathcal U_8 and U9\mathcal U_9 are generated by computer from his hand-made U6\mathcal U_6 and U7\mathcal U_7 (pp. 124--125); the note's own computation packs the even-order trees into H10H_{10} and the odd-order trees into H11H_{11}.

Acceptance. Refereed: the Journal of Combinatorial Mathematics and Combinatorial Computing, volume 8 (zbMATH record read). The site's commentary does not name the note, and the site labels the problem FALSIFIABLE, an open problem, so no reviewed evidence exists. The computation itself has not been repeated, and no independent review is supplied.