Problem 112

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claims/: The 1 claim page of Problem 112, one per claimant's result; the problem's standing derives from them.

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Statement. Let $k=k(n,m)$ be minimal such that any directed graph on $k$ vertices must contain either an independent set of size $n$ or a transitive tournament of size $m$. Determine $k(n,m)$.

Formulation. The site's wording of 2026-09-18 (the page shows no last-edited date). The origin, Erdős and Rado's 1967 paper, states the finite property without graph language: $l_0(m,n)$ is the least $l$ such that every $\{0,1\}$-valued function $\rho$ on the ordered pairs of $\{0,\ldots,l-1\}$ admits either $m$ points $\lambda_0,\ldots,\lambda_{m-1}$ with $\rho(\lambda_i,\lambda_j)=0$ for all $i<j$, or $n$ points with $\rho=1$ in both directions on every pair. Reading $\rho(\lambda,\mu)=0$ as an arc $\lambda\to\mu$, the first alternative is a transitive tournament of size $m$ present as a subgraph and the second an independent set of size $n$, so $k(n,m)=l_0(m,n)$ with the same letters as the site, and the "directed graph" of the statement is an arbitrary binary relation: a pair may carry arcs in both directions or none. The modern literature works with oriented graphs (at most one arc between two vertices) and writes $r(I_n,L_m)$ for the same threshold. The two conventions give the same number (a one-line check made here: deleting one arc from each pair of opposite arcs of a directed graph leaves an oriented graph with the same non-adjacent pairs, whose transitive subtournaments are subgraphs of the original, so the directed threshold is at most the oriented one; oriented graphs are directed graphs, so it is at least). In the oriented setting a transitive tournament of size $m$ as a subgraph is the same as an induced transitive subtournament. The site's second variant, with a directed path in place of the transitive tournament, is a different function (the site gives its value $(n-1)(m-1)$ as an unpublished observation of Hunter and Steiner; under this page's convention, in which $k$ is the least order that forces one of the two structures, that figure is the largest order of a digraph avoiding both, one less than the threshold: $n-1$ disjoint transitive tournaments on $m-1$ vertices avoid both, and the Gallai--Roy theorem gives the matching upper bound) and is not this problem.

Status. Open. No formula for $k(n,m)$, and no determination beyond the values listed in the Current assessment ($k(n,1)=k(1,m)=1$ and $k(n,2)=n$; the tournament column $k(2,m)$ for $m\le6$; $k(3,3)=9$, $k(4,3)=15$, $k(5,3)=23$), was found in the search whose scope the Current assessment records. A pending partial claim, [[problems/ramsey_theory/E0112/claims/2026_09_22_muhamadiev|Muhamadiev's $k(3,4)=21$]], was posted on 22 September 2026. The bounds verified here from the sources read are Erdős and Rado's $k(n,m)\le(2^{m-1}(n-1)^m+n-2)/(2n-3)$ (1967), Larson and Mitchell's $k(n,3)\le n^2$ and their polynomial bound of degree $m-1$ in $n$ with leading coefficient $2^{m-2}/(m-1)!$ (1997), and, for the oriented threshold, $k(n,3)\le n^2-n+3$, $k(n,3)=\Theta(n^2/\log n)$ and $k(n,m)\le2^{17m}n^{m-1}/(\log_2n)^{m-2}$ (Ihringer, Rajendraprasad and Weinert, Discrete Math. 2021). This is a bounded negative finding, not a certificate of openness.

Source. erdosproblems.com/112, accessed 2026-09-18: the problem page (labeled OPEN, with the site's note that no finite computation can settle it; source keys [ErRa67], [LaMi97]; no last-edited date shown; a credit line thanking Zach Hunter and Raphael Steiner), its empty discussion thread and its empty proof-claim tab. Cite as: T. F. Bloom, Erdős Problem #112, https://www.erdosproblems.com/112, accessed 2026-09-18.

References.

• [ErRa67] Erdős, P. and Rado, R., Partition relations and transitivity domains of binary relations. J. London Math. Soc. 42 (1967), 624--633, doi:10.1112/jlms/s1-42.1.624. Theorem 1 (p. 624) and Theorem 2 with relation (3) and Remark (i) (p. 625). Library home: erdos_1967_partition_relations_transitivity_domains_binary_relations.
• [ErRa56] Erdős, P. and Rado, R., A partition calculus in set theory. Bull. Amer. Math. Soc. 62 (1956), 427--489; Theorem 25, which the 1967 paper quotes as its Theorem 1 and which Ihringer, Rajendraprasad and Weinert restate as $r(\omega m,n)=\omega\,r(I_m,L_n)$. Library home: erdos_1956_partition_calculus_set_theory (filed for Problem 1172; its Theorem 25 was not read for this page).
• [LaMi97] Larson, Jean A. and Mitchell, William J., On a problem of Erdős and Rado. Ann. Comb. 1 (1997), 245--252, doi:10.1007/BF02558478. Proposition 3.1 and Lemmas 4.1--4.2 with the corollary $r(K_4^*,L_3)\le16$ (p. 248), Lemma 4.13 with the growth estimates (p. 251), Theorem 2.4 and the table of small values (p. 247). Library home: larson_mitchell_1997_problem_erdos_rado.
• [IRW21] Ihringer, F., Rajendraprasad, D. and Weinert, T., New bounds on the Ramsey number $r(I_m,L_n)$. Discrete Math. 344 (2021), no. 3, 112268, doi:10.1016/j.disc.2020.112268; arXiv:1707.09556 (v3 8 April 2020, 20 pp., whose pagination the locators follow). Theorems 1.1--1.5 (pp. 3--4), Proposition 3.4 (p. 9), Theorem 5.6 (p. 15). Library home: ihringer_2017_new_bounds_ramsey_number_r_i.
• [Be74] Bermond, J.-C., Some Ramsey numbers for directed graphs. Discrete Math. 9 (1974), 313--321, doi:10.1016/0012-365X(74)90077-6. Proposition 2.5 (p. 316), $R(TT_3,K_3^*)=9$, with the definitions of p. 313 and Theorem 2.2 and Proposition 2.4 (pp. 314--315). Library home: bermond_1974_some_ramsey_numbers_directed_graphs.
• [Ba74] Baumgartner, J. E., Improvement of a partition theorem of Erdős and Rado. J. Combin. Theory Ser. A 17 (1974), 134--137, doi:10.1016/0097-3165(74)90037-5. The note's one result, unnumbered (p. 135), $\omega_\alpha\cdot l_0(m,n)\to(m,\omega_\alpha\cdot n)^2$ for all $\alpha$, $m$ and $n$, with the sentences of pp. 134--135 it rests on. Library home: baumgartner_1974_improvement_partition_theorem_erdos_rado.
• [ErMo64] Erdős, P. and Moser, L., On the representation of directed graphs as unions of orderings. Magyar Tud. Akad. Mat. Kutató Int. Közl. 9 (1964), 125--132; Theorem 1 (p. 127), the tournament column. Library home: erdos_1964_representation_directed_graphs_as_unions_orderings.
• [St59] Stearns, R., The voting problem. Amer. Math. Monthly 66 (1959), 761--763. Not held; its tournament-column bound is quoted from [IRW21], p. 2, and its argument is reproduced on p. 126 of [ErMo64].
• [RePa70] Reid, K. B. and Parker, E. T., Disproof of a conjecture of Erdős and Moser on tournaments. J. Combinatorial Theory 9 (1970), 225--238; Theorem 4 (p. 235) with the values of $f(n)$ and the note on a $TT_6$-free $T_{27}$ (pp. 235--236), the tournament column beyond $m=4$. Library home: reid_parker_1970_disproof_conjecture_erdos_moser_tournaments.

Formalization. None found. No file for this problem exists in google-deepmind/formal-conjectures at its revision of 2026-09-18 (main), and the community database (teorth/erdosproblems) at its revision of 2026-09-18 records the problem open (last updated 31 August 2025), not formalized, with no formal proof. The site's "Formalised statement?" indicator reads "No".

Current assessment

The question (site formulation of 2026-09-18). The statement above; OPEN; no last-edited date; no comments and no proof claims. The commentary, restated here, attributes the problem to Erdős and Rado [ErRa67] with their bound $k(n,m)\ll_mn^{m-1}$, in the explicit form $k(n,m)\le(2^{m-1}(n-1)^m+n-2)/(2n-3)$; credits Larson and Mitchell [LaMi97] with a better dependence on $m$, including $k(n,3)\le n^2$; and records an observation it credits to Hunter, that $k(n,m)$ lies between the two-color Ramsey number $R(n,m)$ and the three-color $R(n,m,m)$, so that $k(n,m)\le3^{n+2m}$; it then states the directed-path variant from the graphs problem collection. The sandwich between the two-color and the three-color Ramsey numbers is the site's report of Hunter's observation, not a published source statement, and is not used below. The community database record at the revision linked under Formalization says open and not formalized. On 22 September 2026 the thread received one comment, the computation of $k(3,4)=21$ recorded on its claim page; the proof-claim tab carried no entry on 2026-10-07.

Origin. Erdős and Rado's Theorem 1 (printed p. 624), cited to their 1956 paper's Theorem 25, defines $l_0(m,n)$ by the finite property recorded under Formulation, and the text after it gives $l_0(1,n)=l_0(m,1)=1$ and $l_0(m,2)=2^{m-1}$ for $m\le4$; their Theorem 2 (printed p. 625) gives one integer $l(m,n)$ with $\omega_\alpha l(m,n)\to(m,\omega_\alpha n)^2$ for every initial ordinal $\omega_\alpha$, the least index $l_\alpha(m,n)$ that works for a given $\alpha$ obeying the explicit bound (3), $l_\alpha(m,n)\le(2n-3)^{-1}[2^{m-1}(n-1)^m+n-2]$, which at $\alpha=0$ is the site's bound with the exponent $m$ on $(n-1)$. Remark (i) on the same page conjectures $l_\alpha(m,n)=l_0(m,n)$, which "has so far only been proved when $m\le4$ and $n\le2$"; Baumgartner proved it in 1974: the one result of [Ba74] (p. 135) is $\omega_\alpha\cdot l_0(m,n)\to(m,\omega_\alpha\cdot n)^2$ for all $\alpha$, $m$ and $n$, which with the 1967 paper's negative relation below $\omega_\alpha l_0(m,n)$ gives $l_\alpha(m,n)=l_0(m,n)$, and [IRW21] records it in its Theorem 1.5 as $r(\kappa m,n)=\kappa\,r(I_m,L_n)$ for every infinite initial ordinal $\kappa$, so the whole ordinal family reduces to the finite numbers of this problem. The introduction of the 1967 paper (pp. 624--625) also attributes to Stearns the case $n=2$ in the form of a transitivity theorem, "reproduced in [8; p. 126]", the Erdős--Moser paper.

Exact values (primary sources first). In the letters of the site, $k(n,m)=r(I_n,L_m)$ of [IRW21]. Known exactly:

• $k(n,1)=k(1,m)=1$ and $k(2,m)=2^{m-1}$ for $m\le4$ ([ErRa67], p. 624), and $k(n,2)=n$ ([ErRa67], p. 626, at the start of the proof of Theorem 2, as $l_\alpha(2,n)=n$). The tournament column beyond that: $k(2,5)=14$ and $k(2,6)=28$ ([RePa70], filed as reid_parker_1970_disproof_conjecture_erdos_moser_tournaments and also quoted by [IRW21] p. 2), the column being the inverse of Problem 1216's function. As that card's Bears-on paragraph for this problem states, $k(2,5)=14$ is the paper's $f(14)=5$ with $f(13)=4$ (announced on p. 226, from Theorem 4 and the 13-vertex witness of pp. 235--236), and $k(2,6)=28$ combines the post-submission note's $T_{27}$ with no transitive subtournament on 6 vertices, which gives $k(2,6)>27$, with Corollary 2 at $n=28$, which gives $f(28)\ge6$ (both p. 236); no argument is printed for the note, so that half of the value rests on an unpublished verification by one author. The card records those passages and follows Theorem 4's one-paragraph proof in full. The general bounds on the column are Stearns's $k(2,m)\le2^{m-1}$ and Erdős and Moser's $k(2,m)\ge2^{(m-1)/2}$ (Theorem 1 of [ErMo64]).
• $k(3,3)=9$: Proposition 2.5 of [Be74] (p. 316), stated as $R(TT_3,K_3^*)=9$ in that paper's arc-coloring notation, in which $R(TT_m,K_n^*)$ is this problem's $k(n,m)$ in Erdős and Rado's convention (the card records the translation). The upper bound is the paper's Theorem 2.2 at two colors, $k(3,3)\le R(k(2,3),3)=R(4,3)=9$; the lower bound is the 8-vertex circulant with arcs $i\to j$ for $j-i\equiv2,3\pmod8$, for which the paper asserts the absence of a $TT_3$ without argument and cites Graver and Yackel for the triangle-freeness of the non-adjacency graph; the result page records a one-line check of each. Also quoted by [IRW21] p. 2.
• $k(4,3)=15$ and $k(5,3)=23$: Theorem 1.1 of [IRW21], from Proposition 3.4 and two explicit oriented graphs on 14 and 22 vertices (p. 10; the constructions were not verified here). Acceptance evidence: Discrete Mathematics 344 (2021), a refereed journal (Crossref record accessed); the locators are those of the arXiv v3 of 8 April 2020, whose arXiv comment says that it incorporates reviewers' comments, not compared with the journal text.

No other exact value was found in the sources read or in a refereed source. The paper's Coda (p. 18) names $k(3,4)$ ($r(I_3,L_4)$ in its letters) as the next feasible case; its Proposition 6.1 (p. 18) gives the best small-parameter upper bounds by a recursion, $k(3,4)\le25$ among them. A lead on that case is recorded below.

Bounds (from the sources read). In the site's letters, with $n$ the independent set and $m$ the transitive tournament:

• $k(n,m)\le(2^{m-1}(n-1)^m+n-2)/(2n-3)$ for all $m,n\ge1$ ([ErRa67], relation (3) at $\alpha=0$; a footnote notes the right side is an integer). This is the bound $k(n,m)\ll_mn^{m-1}$ the site states.
• $k(n,3)\le n^2$ for $n\ge2$ (Lemma 4.2 of [LaMi97], p. 248, stated as $r(K_n^*,L_3)\le n^2$ for $n>1$; its three-line induction from Lemma 4.1 was followed here, and [IRW21] Lemma 2.4 attributes the bound to this paper) and $k(n,3)\le n^2-n+3$ for $n\ge2$ (Proposition 3.4, tight for $n\in\{3,4,5\}$ and, the paper says, better than the asymptotic bound for $n\le2^{508}$). For $n=4$ the 1997 paper brackets $14\le k(4,3)\le16$ (its table of p. 247 prints "$14-16$"), the lower half by the 13-vertex digraph of its Proposition 3.1 (p. 248; the printed in-neighborhood of one vertex disagrees with the out-neighborhood columns in one entry, and the digraph the out-neighborhood columns define was checked here by computer to have no transitive triple and no independent 4-set, a filing check, not review), the interval [IRW21] closed at 15.
• $k(n,m)\le2^{m-3}t(n,m)+2^{m-5}\cdot17\binom{n+m-6}{n-2}-\tfrac12$ for $n\ge3$ and $m\ge4$, with $t(n,m)=2\binom{m+n-4}{m-1}+3\binom{m+n-5}{m-2}+\tfrac92\binom{m+n-6}{m-3}$ (Lemma 4.13 of [LaMi97], p. 251; the one-paragraph proof from its Lemma 4.12 was followed, the chain of Lemmas 4.3--4.12 read for structure), a polynomial in $n$ of degree $m-1$ with leading coefficient $2^{m-2}/(m-1)!$ where the Erdős--Rado bound has $2^{m-2}$, and of order $2^{m-5}\cdot17\,m^{n-2}/(n-2)!$ in $m$ where the Erdős--Rado bound grows like $2^{m-1}(n-1)^m/(2n-3)$, that is, $2^m\cdot m^{n-2}$ against $2^m(n-1)^m$ up to factors depending on $n$ alone: the better dependence on $m$ that the site credits to the paper. The paper's recurrence $k(n+1,m+1)\le2k(n+1,m)+k(n,m+1)+1$ for $n>1$ and $m\ge2$ (Lemma 4.3, p. 249, printed without proof after the sentence at the foot of p. 248 that "a similar argument" to that of Lemma 4.1 yields it) gives, in the paper's Maple table (p. 251), the smaller estimate 8,765,184 at $n=m=10$, a figure not reproduced here.
• $k(n,3)=\Theta(n^2/\log n)$ (Theorem 1.2; explicitly $k(n,3)\le508n^2/\log_2n$ by its Corollary 5.2), the same order as the undirected $R(n,3)$, whose lower bound it inherits through $r(I_n,K_3)\le r(I_n,L_3)\le r(I_n,K_4)$ (p. 3). That lower bound is Kim 1995 (the paper's [11]), filed as kim_1995_ramsey_number_has_order_magnitude; its Theorem 1.1 and the consequence $R(3,t)\ge(1/162-o(1))t^2/\log t$ are on pp. 1--2 of the typescript (not the journal's pagination) and paged on theorem_1_1.
• $k(n,m)\le2^{17m}n^{m-1}/(\log_2n)^{m-2}$ for all $n,m\ge2$ (Theorem 5.6, the explicit form of Theorem 1.3, $k(n,m)\le C_mn^{m-1}/(\log n)^{m-2}$ for $m\ge3$), of the same order as the Ajtai--Komlós--Szemerédi bound for $R(n,m)$; it has the same order $n^{m-1}$ as the Erdős--Rado bound and gains the factor $(\log_2n)^{m-2}$ at the cost of the constant $2^{17m}$.

Lower bounds for $m\ge4$ beyond $k(n,m)\ge R(n,m)$ (the underlying graph of a transitive tournament on $m$ vertices is $K_m$, so any orientation of a graph on $R(n,m)-1$ vertices with no $K_m$ and no $n$ independent vertices witnesses this; the site states the same inequality, and [LaMi97] Theorem 2.4, p. 247, records it from Harary and Hell 1974 together with $k(n,m)\le R(n,2^{m-1})$) were not found in the sources read. The problem "Determine $k(n,m)$" therefore stands at: the trivial columns $m\le2$ and the row $n=1$ known exactly, the column $m=3$ known up to a constant factor and exactly for $n\le5$, the column $n=2$ exactly for $m\le6$, and the general case bounded above by $2^{O(m)}n^{m-1}/(\log n)^{m-2}$ with no matching lower bound.

Ordinal side (context). [IRW21] Theorem 1.4 restates the 1956 Theorem 25 as $r(\omega m,n)=\omega\,r(I_m,L_n)$ and Theorem 1.5 (Baumgartner) as $r(\kappa m,n)=\kappa\,r(I_m,L_n)$ for all infinite initial ordinals $\kappa$; the first is second-hand here (the 1956 paper is filed for another problem), and the second is from Baumgartner's note itself ([Ba74]; its statement and the sentences of pp. 134--135 it rests on were read, the proof of pp. 135--137 for structure only), in Erdős and Rado's letters: $\omega_\alpha\cdot l_0(m,n)\to(m,\omega_\alpha\cdot n)^2$ with $m$ the transitive tournament and $n$ the independent set, so $l_\alpha(m,n)=l_0(m,n)=k(n,m)$ for every $\alpha$. Nosal's formulas for $r(\omega^m,n)$, $m\ne4$, mentioned in the Coda, concern a different family.

Pending claim on $k(3,4)$. A comment of 22 September 2026 in the site's discussion thread, by Muhamadiev Faridun with a public repository created the same day, claims $k(3,4)=21$: a circulant witness on 20 vertices, and for the upper bound a case split into 367 cases, each refuted by a SAT solver with a DRAT proof checked by drat-trim, resting on hand lemmas about the neighborhoods of a vertex and a counting lemma. It also claims the brackets $34\le k(4,4)\le50$ and $31\le k(3,5)\le55$. The comment discloses AI assistance (Claude Opus 5). The claim is recorded, unreviewed, on its claim page, and is not a source for any value above.

Search scope. None of the routes below found a formula for $k(n,m)$, a new exact value in a refereed source, or a proof claim.

• The site: problem page, discussion thread and proof-claim tab; the full directory FormalConjectures/ErdosProblems/ (673 entries) of formal-conjectures at the revision linked under Formalization (no file for this problem); the community database at the revision linked there.
• The primary sources read: [ErRa67] printed pp. 624--626, 630 and 632--633; [IRW21] pp. 2--5, 9--11, 15 and 17--18; [ErMo64] printed pp. 125--127.
• arXiv: the API record of 1707.09556 (three versions; no journal reference carried) and the queries abs:"transitive tournament" AND abs:"independent set" (two records, [IRW21] and a 2002 paper on tiling) and abs:"oriented Ramsey" OR abs:"oriented graph" AND abs:"transitive tournament" (21 records, scanned by title: oriented Ramsey numbers of digraphs, tilings and Turán-type problems in oriented graphs, none on $r(I_n,L_m)$).
• Crossref records of [ErRa67], [LaMi97] and the journal version of [IRW21].
• Semantic Scholar: the one record citing [IRW21] (a 2018 paper on polarized partition relations for order types, Q. J. Math.), not on this problem.

Not searched: MathSciNet, zbMATH, Google Scholar, X. Not held: [St59]. The 1956 paper's Theorem 25 is unread, and [RePa70] was not read for this page either. The search above did not include [LaMi97], [Be74] or [Ba74]. [LaMi97] was read for this page at Proposition 3.1, Lemmas 4.1--4.2 with the corollary (p. 248), Lemma 4.13 with the growth estimates (p. 251) and Theorem 2.4 with the table of small values (p. 247), with the short proofs of Lemmas 4.1, 4.2 and 4.13 followed and the chain of Lemmas 4.3--4.12 read for structure. [Be74] was read for this page at its Proposition 2.5 and the passages it rests on (pp. 313--316). [Ba74] was read for this page at its one result and the sentences it rests on (pp. 134--135), with its proof (pp. 135--137) read for structure only. Kim 1995 (Random Structures Algorithms 7 (1995), 173--207) and Alon 1996 (Random Structures Algorithms 9 (1996), 271--278), the [11] and [3] of [IRW21] (reference entries pp. 18--19), are filed as kim_1995_ramsey_number_has_order_magnitude and alon_1996_independence_numbers_locally_sparse_graphs_ramsey. Kim was read for this page at Theorem 1.1 and the $R(3,t)$ consequence (pp. 1--2 of the typescript), paged on theorem_1_1. Alon was read for this page at Proposition 2.1, the independence bound that [IRW21] quotes as its Proposition 5.1 (p. 11) for the upper bound of Theorem 1.2: p. 2 of the preprint (not the journal's pagination); the proposition has no result page of its own, and the Theorem 1.1 it is the main step of is paged on theorem_1_1.

Remaining gaps. (1) In Larson and Mitchell's paper, the site's second source, the 13-vertex witness for $k(4,3)\ge14$ carries a misprinted in-neighborhood entry, and the digraph its out-neighborhood columns define was checked by computer here as a filing check, not review; the proofs of its Lemmas 4.10--4.12 were read for structure only, and its Lemma 4.3 is printed without proof. (2) The lower half of $k(2,6)=28$ rests on the post-submission note of [RePa70], for which the paper prints no argument; the exact value $k(3,3)=9$ rests on Proposition 2.5 of [Be74], whose 8-vertex witness the paper asserts with "It can be shown" and a citation to Graver and Yackel, both checked in one line each on the result page as filing checks, not review. (3) The proofs of [IRW21] are compiled as statements with proof pointers; the two constructions of its Section 4 were not verified. (4) The claimed $k(3,4)=21$ is unreviewed, and its own repository lists what is not machine-checked: the load-bearing counting lemma (273 further cases were never run), the soundness of the per-case symmetry breaking, and the enumeration of the neighborhood blocks, which was not cross-checked against an independent tool; the per-case DRAT proofs were deleted after checking, their digests kept. (5) The identification of the site's "directed graph" with Erdős and Rado's binary relation, and its equivalence with the oriented convention, are checks made here, not statements of the sources.

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.

• graver_yackel_1968_graph_theoretic_results_associated_ramsey_theorem
• baumgartner_1974_improvement_partition_theorem_erdos_rado
• baumgartner_1974_improvement_partition_theorem_erdos_rado / main_theorem
• bermond_1974_some_ramsey_numbers_directed_graphs
• bermond_1974_some_ramsey_numbers_directed_graphs / proposition_2_4
• bermond_1974_some_ramsey_numbers_directed_graphs / proposition_2_5
• bermond_1974_some_ramsey_numbers_directed_graphs / theorem_2_2
• bermond_1974_some_ramsey_numbers_directed_graphs / theorem_3_5
• erdos_1964_representation_directed_graphs_as_unions_orderings
• erdos_1964_representation_directed_graphs_as_unions_orderings / theorem_1
• erdos_1967_partition_relations_transitivity_domains_binary_relations
• erdos_1967_partition_relations_transitivity_domains_binary_relations / theorem_1
• erdos_1967_partition_relations_transitivity_domains_binary_relations / theorem_2
• ihringer_2017_new_bounds_ramsey_number_r_i
• ihringer_2017_new_bounds_ramsey_number_r_i / proposition_3_4
• ihringer_2017_new_bounds_ramsey_number_r_i / theorem_1_1
• ihringer_2017_new_bounds_ramsey_number_r_i / theorem_1_2
• ihringer_2017_new_bounds_ramsey_number_r_i / theorem_5_6
• larson_mitchell_1997_problem_erdos_rado
• larson_mitchell_1997_problem_erdos_rado / lemma_4_13
• larson_mitchell_1997_problem_erdos_rado / lemma_4_2
• larson_mitchell_1997_problem_erdos_rado / lemma_4_4
• larson_mitchell_1997_problem_erdos_rado / proposition_3_1
• mccarthy_2025_mathon_type_construction_digraphs_improved_lower_bounds
• mccarthy_2025_mathon_type_construction_digraphs_improved_lower_bounds / corollary_8
• mccarthy_2025_mathon_type_construction_digraphs_improved_lower_bounds / theorem_1
• mccarthy_2025_mathon_type_construction_digraphs_improved_lower_bounds / theorem_7
• reid_parker_1970_disproof_conjecture_erdos_moser_tournaments
• reid_parker_1970_disproof_conjecture_erdos_moser_tournaments / corollary_2
• reid_parker_1970_disproof_conjecture_erdos_moser_tournaments / theorem_4
• erdos_1956_partition_calculus_set_theory