Problem 743

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claims/: The 6 claim pages of Problem 743, one per claimant's result; the problem's standing derives from them.

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Statement. Let $T_2,\ldots,T_n$ be a collection of trees such that $T_k$ has $k$ vertices. Can we always write $K_n$ as the edge disjoint union of the $T_k$?

Formulation. The site's wording, accessed (the page shows no last-edited date). This is the tree packing conjecture of Gyárfás (stated with Lehel in the 1976 Keszthely proceedings, published 1978), for every $n\ge2$: since $\sum_{k=2}^n(k-1)=\binom n2$, a packing of edge-disjoint copies of $T_2,\ldots,T_n$ into $K_n$ uses every edge, so "edge disjoint union" and "packing" ask the same thing; the sources index the trees $T_1,\ldots,T_n$ with $|T_i|=i$, $T_1$ having no edges. For a single $n$ the question is a finite check. The label FALSIFIABLE is the site's; the site's page defines it as an open problem that a finite counterexample could refute. Erdős's words are quoted under the Current assessment.

Status. Falsifiable is the site's label; the conjecture is open, and that a single $n$ with a family of trees that does not pack would refute it is a note of the Current assessment, not a claim. No proof, disproof or accepted proof claim was found in the search whose scope the Current assessment records: the one arXiv proof claim (Chalise, Clark and Gnang, arXiv:2410.13840, 2024) was withdrawn by its authors on 1 September 2026, citing an error in the proof of their Composition Lemma 3.10 (claim page, withdrawn). The results that settle instances of the conjecture are recorded as partial claim pages, accepted or pending as their evidence allows, and with the withdrawn page they are the pages from which the frontmatter standing, open, is derived; no full claim is pending. What is proved: the conjecture for all large $n$ when the trees beyond the first $\varepsilon n$ have bounded maximum degree (Joos, Kim, Kühn and Osthus, Theorem 1.2, J. Eur. Math. Soc. 21 (2019), refereed; the accepted partial claim Joos, Kim, Kühn and Osthus 2016/2019), for all large $n$ when every tree has maximum degree at most $cn/\log n$ (Allen, Böttcher, Clemens, Hladký, Piguet and Taraz, Theorem 6, arXiv 2021--2026, a preprint with no journal record; the pending partial claim Allen, Böttcher, Clemens, Hladký, Piguet and Taraz 2021), for the largest $\varepsilon n$ trees of any degrees for every $n$ (Janzer and Montgomery, Theorem 1.2, arXiv v2 of April 2026, "accepted for publication" per its arXiv record), for the star and path families of Gyárfás and Lehel (a 1978 proceedings paper, not held; the pending partial claim Gyárfás and Lehel 1978), and by computer for every $n\le11$ (Guichard and Massman 1990, J. Combin. Math. Combin. Comput., the accepted partial claim Guichard and Massman 1990; Fishburn's $n\le9$ before it, J. Graph Theory 7 (1983), refereed, not held, the accepted partial claim Fishburn 1983). A forum computation of 8 September 2026 reports $n=12$; it is a lead. This is a bounded negative finding, not a certificate of openness.

Source. erdosproblems.com/743, accessed 2026-09-19: the problem page (FALSIFIABLE, with the site's definition of the label; no last-edited date shown; source key [Er81]; commentary citing [GyLe78], [Fi83], [Bo83], [JKKO19], [ABCHPT21] and [JaMo24]; a thanks line; "Formalised statement? No"), its six-comment discussion thread (28 February 2026 to 8 September 2026) and its empty proof-claim tab. Cite as: T. F. Bloom, Erdős Problem #743, https://www.erdosproblems.com/743, accessed 2026-09-19.

References.

• [JKKO19] Joos, Felix and Kim, Jaehoon and Kühn, Daniela and Osthus, Deryk, Optimal packings of bounded degree trees. J. Eur. Math. Soc. 21 (2019), no. 12, 3573--3647, doi:10.4171/JEMS/909 (issued 5 August 2019; Crossref record accessed). Paged in arXiv:1606.03953v2 (13 March 2019, 56 pp., "final version (December 2017)" per the arXiv record); Conjecture 1.1, p. 1; Theorem 1.2, p. 2. The journal text is not held; the page numbers are the arXiv version's. Library home: joos_2019_optimal_packings_bounded_degree_trees; paged at theorem_1_2.
• [ABCHPT21] Allen, Peter and Böttcher, Julia and Clemens, Dennis and Hladký, Jan and Piguet, Diana and Taraz, Anusch, The tree packing conjecture for trees of almost linear maximum degree. arXiv:2106.11720 (v1 22 June 2021; v2 20 June 2022, 157 pp., the version paged here; v3 8 September 2026, 179 pp., not held); no journal reference in the arXiv record. Conjecture 2, p. 3; Theorems 5--8, p. 5 (v2). Library home: allen_2021_tree_packing_conjecture_trees_almost_linear; paged at theorem_6.
• [JaMo24] Janzer, Barnabás and Montgomery, Richard, Packing the largest trees in the tree packing conjecture. arXiv:2403.10515 (v1 15 March 2024; v2 9 April 2026, 34 pp., the version paged here, "Version accepted for publication", no journal named). Conjecture 1.1 and the history, p. 1; Theorem 1.2, p. 2. Library home: janzer_2024_packing_largest_trees_tree_packing_conjecture; paged at theorem_1_2.
• [GuMa90] Guichard, David R. and Massman, John D., A note on packing complete graphs with trees. J. Combin. Math. Combin. Comput. 8 (1990), 123--126 (zbMATH record 22649; no DOI). Not a site key; named in the thread on 27 August 2026. The note is open access on the publisher's site. Library home: guichard_1990_note_packing_complete_graphs_trees; paged at verification_p124.
• [Fi83] Fishburn, P. C., Balanced integer arrays: a matrix packing theorem. J. Combin. Theory Ser. A 34 (1983), no. 1, 98--101, doi:10.1016/0097-3165(83)90045-6 (Crossref record accessed; text accessed). The site's key. Proposition 1, p. 98, with Graham's observation, p. 99: the degree sequences of $T_1,\ldots,T_n$ pack into that of $K_n$ (Graham's conjecture), as [JKKO19] (p. 1) and [GuMa90] (p. 123) cite it; the note states no result on the conjecture itself and no $n\le9$ verification. Library home: fishburn_1983_balanced_integer_arrays_matrix_packing_theorem; paged at proposition_1.
• [Fi83b] Fishburn, P. C., Packing graphs with odd and even trees. J. Graph Theory 7 (1983), no. 3, 369--383, doi:10.1002/jgt.3190070309 (Crossref record accessed, whose abstract ends "The conjectures are also valid for all trees when $n\le9$, so that the Gyárfás-Lehel conjecture holds for $n\le9$"). Not held; not a site key; the paper [JaMo24] and [GuMa90] cite for the $n\le9$ verification.
• [GyLe78] Gyárfás, A. and Lehel, J., Packing trees of different order into $K_n$. Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Colloq. Math. Soc. János Bolyai 18, North-Holland (1978), 463--469 (as [GuMa90] and [ABCHPT21] cite it; the site's reference text gives no venue). Not held (print only; no attempt); its cases (all but at most two trees stars, or all stars or paths) are quoted from the site and [JaMo24]; [Bo83] (p. 203) attests the stars case.
• [Bo83] Bollobás, Béla, Some remarks on packing trees. Discrete Math. 46 (1983), no. 2, 203--204, doi:10.1016/0012-365X(83)90254-6 (Crossref and zbMATH records accessed; the publisher's open-archive license dated 2013; text accessed). The Theorem, p. 203; the Erdős--Sós remark, p. 204. Library home: bollobas_1983_some_remarks_packing_trees; paged at theorem_p203.
• [CCG24] Chalise, Parikshit and Clark, Antwan and Gnang, Edinah K., A Proof of the Tree Packing Conjecture. arXiv:2410.13840 (v1 17 October 2024; v2 23 October 2024; v3 1 September 2026, withdrawn). A withdrawn claim, not a source (arXiv record); recorded as the claim page chalise_clark_gnang.
• [Er81] Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica 1 (1981), no. 1, 25--42, doi:10.1007/BF02579174; p. 15 of the retyped copy, which has its own pagination. Library home: erdos_1981_combinatorial_problems_which_i_would_most.

Formalization. None. No file ErdosProblems/743.lean exists in formal-conjectures neither in the directory FormalConjectures/ErdosProblems/ (682 entries) nor elsewhere in the tree (1,751 entries); the site's page shows "Formalised statement? No (create one)"; the community database (teorth/erdosproblems, data/problems.yaml) records the problem falsifiable (last update 31 August 2025), unformalized, with no formalized statement and the comment "tree packing conjecture".

Current assessment

The question (site formulation accessed). The statement above; FALSIFIABLE, which the site defines as open but refutable by a finite counterexample; no last-edited date. The commentary, in this page's words: the site calls this Gyárfás's tree packing conjecture and lists the known cases, each with its key: [GyLe78] for families in which every tree but at most two is a star, and for families of stars and paths; [Fi83] for $n\le9$; [Bo83] for a greedy packing of the smallest $\lfloor n/\sqrt2\rfloor$ trees; [JKKO19] for bounded maximum degree; [ABCHPT21] for maximum degree $O(n/\log n)$; and [JaMo24] for a linear number of the largest trees. The six thread comments and the empty proof-claim tab are recorded below. The community database lists the problem as falsifiable as of its last update, 31 August 2025.

The origin. [Er81], p. 15 of the copy: "Another attractive conjecture of Gyárfás states that if $T_k$, $k=2,3,\ldots,n$ is any set of $n-1$ trees, and $v(T_k)=k$, then $K(n)$ is the edge disjoint union of the $T_k$'s", stated without a reference. The three modern sources state it as Conjecture 1.1 ([JKKO19], p. 1, "Gyárfás and Lehel [19]"; [JaMo24], p. 1, "Gyárfás (see [9]) from 1976") and Conjecture 2 ([ABCHPT21], p. 3, "Gyárfás [11] formulated this conjecture in 1978"), the 1976 colloquium having been published in 1978.

What is proved, by regime.

• Bounded degree, large $n$: Theorem 1.2 of [JKKO19] (arXiv v2 p. 2): "For all $\Delta\in\mathbb N$, there are $N\in\mathbb N$ and $\varepsilon>0$ such that for all $n\ge N$ the following holds. Suppose that for each $i\in[n]$, we have a tree $T_i$ with $|T_i|=i$ and suppose $\Delta(T_i)\le\Delta$ for all $i>\varepsilon n$. Then $K_n$ decomposes into $T_1,\ldots,T_n$." Refereed (J. Eur. Math. Soc. 21 (2019), issue 12). The first $\varepsilon n$ trees may have any degrees.
• Maximum degree $O(n/\log n)$, large $n$: Theorem 6 of [ABCHPT21] (v2 p. 5): "There exist $c>0$ and $n_0\in\mathbb N$ such that for each $n>n_0$ any family of trees $(T_s)_{s\in[n]}$ with $v(T_s)=s$ and $\Delta(T_s)\le\frac{cn}{\log n}$ packs into $K_n$." Deduced, through its Theorem 8, from the paper's Theorem 10 and an earlier perfect packing theorem for degenerate graphs with many leaves (its Theorem 5, Theorem 2 of Allen, Böttcher, Clemens and Taraz); the paper (p. 6) says the degree condition of Theorem 8 is optimal for quasirandom hosts, which does not bound the degrees allowed in $K_n$. A preprint: a v3 of 8 September 2026 exists and was not compared; recorded with the preprint qualification, not compiled as accepted.
• The largest $\varepsilon n$ trees, every $n$, no degree bound: Theorem 1.2 of [JaMo24] (v2 p. 2): "There exists a constant $\varepsilon>0$ such that the following holds with $r=\varepsilon n$ for all $n$. If $T_{n-r+1},\ldots,T_n$ are trees with $|T_i|=i$ for each $n-r<i\le n$, then $T_{n-r+1},\ldots,T_n$ pack into $K_n$." It proves Bollobás's 1995 conjecture that the largest $k$ trees pack for large $n$ (known for $k\le3$ by Hobbs, Bourgeois and Kasiraj and $k\le5$ by Żak, p. 1). The arXiv record calls v2 "Version accepted for publication" and names no journal; recorded with that qualification.
• The smallest trees: the Theorem of [Bo83] (p. 203): "Suppose $3\le s<\frac12\sqrt2\,n$ and $T_2,T_3,\ldots,T_s$ are trees such that $T_i$ has order $i$ for each $i$. Then for every $k$, $2\le k<s$, every packing of $T_{k+1},T_{k+2},\ldots,T_s$ into $K^n$ can be extended to a packing of $T_k,T_{k+1},\ldots,T_s$ into $K^n$. In particular, $T_2,T_3,\ldots,T_s$ can be packed into $K^n$." Since $\sqrt2$ is irrational this packs the smallest $\lfloor n/\sqrt2\rfloor$ trees (counting $T_1$) greedily in descending order of size, for $n\ge5$; its proof occupies one page. The note's closing remark (p. 204) is that the Erdős--Sós conjecture "would allow one to replace the bound $\frac12\sqrt2\,n$ in the theorem by $\frac12\sqrt3\,n$, which would be essentially best possible" (the $\lfloor\sqrt3n/2\rfloor$ that [JaMo24] prints, p. 1); no argument is printed for the remark. The note's introduction (p. 203) also attests Gyárfás and Lehel's proof of the case where all but at most two trees are stars, and Straight's verification of the conjecture for $n\le7$ (Topics in Graph Theory, N.Y. Acad. Sci., 1979, 190--192; not held). Balogh and Palmer (2013) pack, for large $n$, the next $\frac1{10}n^{1/4}$ largest trees when the largest is skipped, or the largest $\frac1{10}n^{1/4}$ when none is a star ([JaMo24] pp. 1--2; [JKKO19] p. 1 states it as packing $T_{n-n^{1/4}/10},\ldots,T_{n-1}$ into $K_n$).
• Special classes: every tree but at most two a star, or every tree a star or a path ([GyLe78], per the site and [JaMo24] p. 1); stars or double stars (Hobbs, per [JaMo24]); Fishburn's classes (per [GuMa90] pp. 123--124).
• Small $n$: every $n\le9$, by hand (Fishburn produced $\mathcal U_2$ through $\mathcal U_7$ by hand and then verified $H_8\in\mathcal U_8$ and $H_9\in\mathcal U_9$, per [GuMa90] p. 124; [Fi83b]: the Crossref abstract, ends "so that the Gyárfás-Lehel conjecture holds for $n\le9$"; attested by [JaMo24] p. 1 and [GuMa90] p. 123), and, by computer, $n=10$ and $n=11$: Guichard and Massman (p. 124) generated Fishburn's universally recursive families $\mathcal U_8$ and $\mathcal U_9$ by computer and showed directly that the one exceptional tree in each case still packs, "proving the Gyárfás--Lehel conjecture for $n=10$" and "for $n=11$" (abstract: "Using a computer, we have shown that the conjecture is true through $n=11$ but also that an approach suggested by Fishburn is unlikely to work in general"). This is the published frontier of the finite check.

A site-versus-source item. The site attributes the $n\le9$ verification to [Fi83], "Balanced integer arrays: a matrix packing theorem" (J. Combin. Theory Ser. A 34 (1983), 98--101). [JKKO19] (p. 1) cites that note for a different fact, that the degree sequences of $T_1,\ldots,T_n$ can be "packed" into the degree sequence of $K_n$; [JaMo24] (p. 1) cites Fishburn's other 1983 paper, "Packing graphs with odd and even trees" (J. Graph Theory 7 (1983), 369--383), for $n\le9$; [GuMa90] cites the two papers jointly on p. 123 ("Fishburn [2,3]") and on p. 124 attributes to its reference [3], that paper, the universally recursive families from which $n\le9$ follows; and that paper's abstract (Crossref record) states the $n\le9$ result. [Fi83], a four-page note, proves Proposition 1 (p. 98), "If $n\ge1$ and $t^i$ for $i=1,\ldots,n$, is a vector of $i$ positive integers that sum to $2i-1$, then there is an $n\times n$ nonnegative matrix each of whose columns sums to $n$ such that the nonzero entries in row $i$ are a permutation of the components of $t^i$", which with Graham's observation (p. 99) is the degree-sequence packing [JKKO19] cites, and it states no result on the conjecture itself for any $n$. The site's key for the $n\le9$ verification therefore names the wrong Fishburn paper; the result belongs to [Fi83b], which is not held, and its attestation stays second-hand. The difference concerns the key, not the mathematics.

Which results are claims. The results that settle instances of the conjecture have partial claim pages (linked under Status): [JKKO19], [ABCHPT21], Fishburn's $n\le9$ ([Fi83b]), [GyLe78] and [GuMa90]. [Bo83] and [JaMo24] pack only some of the trees of a family, the smallest $\lfloor n/\sqrt2\rfloor$ and the largest $\varepsilon n$, and settle no instance of the conjecture, so they are known results and not claims.

The withdrawn proof claim and the thread (leads with provenance). The arXiv claim has a claim page; the thread's computation reports are posts with code repositories and no manuscript, so they get no claim page and are recorded on this page.

• [CCG24], A Proof of the Tree Packing Conjecture (arXiv:2410.13840, v1 17 October 2024, v2 23 October 2024), whose abstract claims the conjecture for all $n$ by a polynomial method and a "complete labeling" reformulation; recorded as the withdrawn claim page Chalise, Clark and Gnang 2024. Its arXiv record, shows v3 of 1 September 2026 with the comment "Withdrawn due to an error in the proof of Lemma 3.10 (Composition Lemma), which was based on the argument in arXiv:2202.03178v2". The thread recorded the claim before the withdrawal: a comment of 28 February 2026 outlines the paper (augmented functional trees, a polynomial certificate $P_g(X,y)=V(X)E_g(X,y)$, Proposition 3.4, the Composition Lemma 3.10, Theorem 1.9), notes that no refereed version or independent verification was found, and points to a MathOverflow discussion of claimed proofs of graph labeling conjectures; a comment of 1 March 2026 reports that people who tried to verify the paper did not trust its arguments; a second comment of 1 March 2026 reports a GPT-based check, as its poster names it, whose output claimed two major gaps, with the poster's caveat that such claims are not guaranteed. As of 2026-09-19 the site's page did not mention the claim or the withdrawal, and its label was unchanged; the authors withdrew the claim on 1 September 2026.
• Verification at $n=10$ (15 August 2026, the account RajveerKapoor): the poster reports, with the help of an AI system the post names as Opus 5, that the conjecture holds for $n=10$, all $45{,}376{,}056$ sequences $(T_2,\dots,T_{10})$ with $|T_k|=k$ decomposing $K_{10}$ (largest tree first, memoized residual graphs by isomorphism class; about 150 seconds on one core; memo hits rechecked against an independent isomorphism test; restricted subfamilies only at $n=11$); the post links a repository (github.com/RajveerKapoor/erdos-work, whose head commit on 2026-09-19 was of 14 August 2026; not read). The named system is the poster's own provenance statement. This recomputes a case [GuMa90] settled in 1990, as the next comment noted.
• The [GuMa90] pointer (27 August 2026, the account pawelkwaczynski): quotes the note's abstract, "Using a computer, we have shown that the conjecture is true through $n=11$, but also that an approach suggested by Fishburn is unlikely to work in general", and concludes that $n=12$ is the first case no published exhaustive check covers. The note is open access (see References).
• Verification at $n=12$ (8 September 2026, the same account): the poster reports that every sequence $(T_2,\dots,T_{12})$ with $|T_k|=k$ packs into $K_{12}$ (the 551 trees on 12 vertices fixed one at a time, trees placed largest first, residual graphs memoized by canonical form; about 220 core-hours; $n\le11$ rederived, with $n=11$ taking 290 CPU-seconds and agreeing with [GuMa90]; a second implementation agrees only up to $n=9$), and says that $n=12$ rests on one implementation and should not be called independently verified until someone else's code reproduces it; the post links github.com/pawelkwaczynski/erdos743 (head commit on 2026-09-19 of 8 September 2026; not read). A computation claim in its poster's words, unreviewed; the published frontier stays $n\le11$. The reactions on the page (one account marked "Currently working on") are not evidence of anything.

Search scope. None of the routes below found a proof, a disproof, an accepted proof claim, or a refereed verification beyond $n\le11$.

• The site: problem page, discussion thread and proof-claim tab; the formal-conjectures directory and tree (no file 743); the community database entry.
• arXiv API: the records of 2106.11720 (v3 of 8 September 2026 is the latest), 2403.10515 (v2 of 9 April 2026), 1606.03953 (v2 of 13 March 2019) and 2410.13840 (v3 withdrawn 1 September 2026); the search abs:"tree packing conjecture" sorted by date (twelve records, by title: [JaMo24], the approximate versions of 2014--2019, [JKKO19], and packing papers of 2011--2021; nothing newer than [JaMo24] and no new proof).
• Crossref: the records of [JKKO19] (by bibliographic query), [Bo83] (by query), [Fi83], [Fi83b] (by query, with its abstract), [Er81]; queries for [GuMa90] and [GyLe78] returned no record for either. zbMATH Open: the records of [GuMa90] (with its summary), [Bo83] and [Fi83b].
• The publisher's article page for [GuMa90] (HTTP 200) and its PDF link (HTTP 200, application/pdf); one paced request to the publisher's PDF endpoint for [Bo83] (HTTP 403, a challenge page).
• GitHub API: the head commits of the two repositories the thread links (records only; no file read). The thread's chat-transcript link is not cited (a rule of this repository), and its MathOverflow pointer was not opened.
• The primary sources at the pages cited: [JKKO19] pp. 1--2, [ABCHPT21] pp. 3 and 5, [JaMo24] pp. 1--2, [GuMa90] pp. 123--126 and [Er81] copy p. 15.

Not searched: MathSciNet, Google Scholar, X; Semantic Scholar (its citation endpoint for 2410.13840 answered HTTP 429 and was not retried). Not held: [GyLe78], [Fi83b], Pritikin's preprint, Straight's 1979 note, the Hobbs--Bourgeois--Kasiraj, Żak and Balogh--Palmer papers, the journal version of [JKKO19], v3 of [ABCHPT21], the published [JaMo24].

Remaining gaps. (1) The conjecture is open for general $n$; the proved regimes are bounded degree, degree $O(n/\log n)$, the largest $\varepsilon n$ trees, special classes and $n\le11$. (2) [ABCHPT21] and [JaMo24] are compiled as preprints with their qualifications; reopening condition for the qualification: a journal record for either. (3) The $n=12$ computation is an unreviewed forum claim; a second implementation reproducing it, or a refereed report, would extend the frontier. (4) [Fi83b] and [GyLe78] are not held; their results are second-hand from the other sources and the site. [Fi83] proves Graham's degree-sequence packing and contains no verification of the conjecture, so the site's key for the $n\le9$ result is recorded as naming the wrong Fishburn paper. (5) Proof coverage: statements only for [JKKO19], [ABCHPT21], [JaMo24] and [GuMa90]; the one-page proofs of [Bo83] and [Fi83] are followed in full; nothing is independently reviewed. (6) The label FALSIFIABLE is recorded as the site's; the problem's standing is open, derived from its claim pages, and that a finite counterexample would refute the conjecture is this body note, not a claim.

Known results

• Joos--Kim--Kühn--Osthus, Theorem 1.2 (2019, refereed): bounded degree beyond the first $\varepsilon n$ trees, large $n$.
• Allen--Böttcher--Clemens--Hladký--Piguet--Taraz, Theorem 6 (preprint, 2022 version paged): maximum degree at most $cn/\log n$, large $n$.
• Janzer--Montgomery, Theorem 1.2 (accepted preprint, 2026 version paged): the largest $\varepsilon n$ trees pack for every $n$.
• Guichard--Massman 1990 (1990, open access): true for $n=10$ and $n=11$ by computer; Fishburn ([Fi83b], not held): true for $n\le9$.
• Bollobás 1983, Theorem (1983): the smallest $\lfloor n/\sqrt2\rfloor$ trees pack greedily, any packing of the larger ones among them extending by the next smaller.
• Fishburn 1983, Proposition 1 (1983): Graham's necessary condition, that the degree sequences of $T_1,\ldots,T_n$ pack into that of $K_n$, holds for every $n$; not a packing of the trees, and not the $n\le9$ verification the site's key suggests.
• [GyLe78] (not held; per the site and [JaMo24]): stars and paths.
• [Er81], p. 15: the conjecture in Erdős's words.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• allen_2021_tree_packing_conjecture_trees_almost_linear
• allen_2021_tree_packing_conjecture_trees_almost_linear / theorem_10
• allen_2021_tree_packing_conjecture_trees_almost_linear / theorem_6
• allen_2021_tree_packing_conjecture_trees_almost_linear / theorem_8
• bollobas_1983_some_remarks_packing_trees
• bollobas_1983_some_remarks_packing_trees / theorem_p203
• fishburn_1983_balanced_integer_arrays_matrix_packing_theorem
• fishburn_1983_balanced_integer_arrays_matrix_packing_theorem / proposition_1
• fishburn_1983_balanced_integer_arrays_matrix_packing_theorem / theorem_1
• guichard_1990_note_packing_complete_graphs_trees
• guichard_1990_note_packing_complete_graphs_trees / verification_p124
• gyarfas_2023_problems_close_my_heart
• gyarfas_2023_problems_close_my_heart / conjecture_3_1
• janzer_2024_packing_largest_trees_tree_packing_conjecture
• janzer_2024_packing_largest_trees_tree_packing_conjecture / theorem_1_2
• joos_2019_optimal_packings_bounded_degree_trees
• joos_2019_optimal_packings_bounded_degree_trees / theorem_1_2
• joos_2019_optimal_packings_bounded_degree_trees / theorem_1_3
• joos_2019_optimal_packings_bounded_degree_trees / theorem_1_7
• erdos_1981_combinatorial_problems_which_i_would_most