Problem 165

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Statement. Give an asymptotic formula for $R(3,k)$.

Formulation. The site's wording (page last edited 7 March 2026). $R(3,k)$ is the least $n$ such that every graph on $n$ vertices contains a triangle or an independent set of size $k$; the sources also write $r(3,k)$, $r(C_3,K_k)$ and $f(3,k)$. An asymptotic formula means a function $f$ with $R(3,k)=(1+o(1))f(k)$. The order of magnitude has been known since 1995, $R(3,k)=\Theta(k^2/\log k)$ with natural logarithms, so the question is the constant $c$ in $R(3,k)\sim c\,k^2/\log k$, if the limit exists. The conjectured formula is $R(3,k)=(\frac12+o(1))k^2/\log k$ (Campos, Jenssen, Michelen and Sahasrabudhe 2025, restated by Hefty, Horn, King and Pfender).

Status. Open, the site's label. No asymptotic formula is proved; the search whose scope the Current assessment records found none, and no proof claim exists. The known bounds, each checked against its source, are

$$\Bigl(\frac12+o(1)\Bigr)\frac{k^2}{\log k}\ \le\ R(3,k)\ \le\ (1+o(1))\frac{k^2}{\log k},$$

the lower bound Hefty, Horn, King and Pfender's Theorem 1.2 (arXiv preprint, v3 of February 2026, no journal record), after Kim's $1/162$ (1995, refereed), Bohman and Keevash's and Fiz Pontiveros, Griffiths and Morris's $1/4$ (refereed, 2021 and 2020) and Campos, Jenssen, Michelen and Sahasrabudhe's $1/3$ (preprint, 2025); the upper bound Shearer's (1983, refereed; Theorem 1, the independence bound $\alpha\ge n(d\ln d-d+1)/(d-1)^2$ for triangle-free graphs of average degree $d$, from which the Ramsey bound follows by an elementary step), sharpening Ajtai, Komlós and Szemerédi (1980, refereed; Theorem 3, $R(3,x)<100x^2/\ln x$, from their Theorem 2, $\alpha(G)\ge0.01(n/t)\ln t$ for triangle-free graphs of average degree $t$). Campos, Jenssen, Michelen and Sahasrabudhe conjecture $c=1/2$, and Hefty, Horn, King and Pfender restate the conjecture as their Conjecture 1.1 and support it. This is a bounded negative finding, not a certificate of openness.

Source. erdosproblems.com/165, accessed 2026-09-18: the problem page (labeled OPEN, with the site's note that no finite computation can settle it; prize offered; last edited 7 March 2026; source keys [Er61], [Er71], [Er78, p. 34], [Er90b], [Er93, p. 339], [Er97c]; commentary citing [Ki95], [Sh83], [AKS80], [BoKe21], [PGM20], [CJMS25], [HHKP25] and Problems 544, 986 and 1013; with the site's thanks to two commenters), its five-comment discussion thread (1 October 2025 to 10 June 2026) and its empty proof-claim tab. Cite as: T. F. Bloom, Erdős Problem #165, https://www.erdosproblems.com/165, accessed 2026-09-18.

References.

• [HHKP25] Hefty, Z., Horn, P., King, D. and Pfender, F., Improving $R(3,k)$ in just two bites. arXiv:2510.19718 (v1 22 October 2025; v3 19 February 2026, 18 pages). Preprint; no journal record. Theorem 1.2, Theorem 1.3 and Conjecture 1.1, p. 2. Library home: hefty_2025_improving_just_two_bites.
• [CJMS25] Campos, M., Jenssen, M., Michelen, M. and Sahasrabudhe, J., A new lower bound for the Ramsey numbers $R(3,k)$. arXiv:2505.13371 (v1 19 May 2025, 52 pages). Preprint; no journal record. Theorem 1.1 and display (1), pp. 1--2; Conjecture 1.2, p. 4. Library home: campos_2025_new_lower_bound_ramsey_numbers.
• [BoKe21] Bohman, T. and Keevash, P., Dynamic concentration of the triangle-free process. Random Structures Algorithms 58 (2021), no. 2, 221--293, DOI 10.1002/rsa.20973; arXiv:1302.5963 (v2 4 September 2019, the version read, 75 pages; no file held; its pagination is not the journal's). Theorem 1.3, p. 2. Library home: bohman_2021_dynamic_concentration_triangle_free_process.
• [PGM20] Fiz Pontiveros, G., Griffiths, S. and Morris, R., The triangle-free process and the Ramsey number $R(3,k)$. Mem. Amer. Math. Soc. 263 (2020), no. 1274, v+125 pp., DOI 10.1090/memo/1274; arXiv:1302.6279 (v2 24 March 2018, 154 pages; its pagination is not the Memoir's). Theorem 1.2 and Conjecture 1.3, p. 4. Library home: fizpontiveros_2020_triangle_free_process_ramsey_number.
• [Ki95] Kim, J. H., The Ramsey number $R(3,t)$ has order of magnitude $t^2/\log t$. Random Structures Algorithms 7 (1995), no. 3, 173--207, DOI 10.1002/rsa.3240070302; cited by the pages of a 36-page typescript without the journal pagination. Theorem 1.1, typescript p. 1; the $R(3,t)$ bound, p. 2. Library home: kim_1995_ramsey_number_has_order_magnitude.
• [Sh83] Shearer, J. B., A note on the independence number of triangle-free graphs. Discrete Math. 46 (1983), no. 1, 83--87, DOI 10.1016/0012-365X(83)90273-X. Theorem 1 and its proof, printed pp. 83--84; the paper prints the independence bound, not the Ramsey bound, and the step between them is recorded on the result page. Library home: shearer_1983_note_independence_number_triangle_free_graphs; paged at theorem_1.
• [AKS80] Ajtai, M., Komlós, J. and Szemerédi, E., A note on Ramsey numbers. J. Combin. Theory Ser. A 29 (1980), no. 3, 354--360, DOI 10.1016/0097-3165(80)90030-8. Theorem 2, printed p. 355, and Theorem 3 with its proof, printed p. 358; the introduction's account of the earlier bounds, p. 354. Library home: ajtai_1980_note_ramsey_numbers; paged at theorem_2 and theorem_3.
• [Mo26] Morris, R., Some recent results in Ramsey theory. Proc. ICM 2026, Vol. 2, 210--239, DOI 10.1137/25m1833369 (published online 13 July 2026); arXiv:2601.05221 (v1 8 January 2026, 37 pages). Theorem 1.2, p. 2. Expert attestation, not a review. Library home: morris_2026_recent_results_ramsey_theory.
• [GrYa68] Graver, J. E. and Yackel, J., Some graph theoretic results associated with Ramsey's theorem. J. Combinatorial Theory 4 (1968), 125--175; Proposition 9, printed p. 154: the 1968 upper bound $R(3,k)\ll k^2\log\log k/\log k$ in the paper's convention, one less than the usual Ramsey number. Library home: Proposition 9.
• [Er71] Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969), Academic Press (1971), 97--109; item 6, printed pp. 98--99. Library home: erdos_1971_unsolved_problems_graph_theory_combinatorial_analysis.
• [Er78] Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Proc. Ninth Southeastern Conf. (Boca Raton, 1978), Congressus Numerantium XXI (1978), 29--40; display (4) and the sentence after it, printed p. 34. Library home: erdos_1978_problems_results_combinatorial_analysis_combinatorial_number.
• [Er81c] Erdős, P., Some new problems and results in graph theory and other branches of combinatorial mathematics. Lecture Notes in Math. 885 (1981), 9--17; item (5), printed p. 10, and the closing sentence of p. 11. Not a site key for this problem; the site's key for Problem 544. Library home: erdos_1981_new_problems_results_graph_theory_other.
• [Er61] Erdős, P., Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961), 221--254. The passage behind the site's key is Part II, item 4, printed pp. 240--241: with $f(2,k,l)$ the two-class Ramsey function of p. 240, p. 241 reads "It would be interesting to determine $f(i,k,l)$ explicitely [sic], this seems very difficult even for $i=2$" and "I can prove that (II.4.2) $f(2,3,k)>ck^2/(\log k)^2$ but could not decide whether $f(2,3,k)>c_2k^2$ is true". Library home: erdos_1961_unsolved_problems.
• [Er90b] Erdős, P., Problems and results on graphs and hypergraphs: similarities and differences. Mathematics of Ramsey theory, Algorithms Combin. 5, Springer (1990), 12--28; display (13) and the asymptotic-formula sentence, printed p. 18. Library home: erdos_1990_problems_results_graphs_hypergraphs_similarities_differences.
• [Er93] Erdős, P., Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. 16 (1993), 333--350; Chapter II, displays (7) and (8) and the asymptotic-formula sentence, printed pp. 338--339. The site cites p. 339. Library home: erdos_1993_my_favorite_solved_unsolved_problems_graph_theory.
• [Er97c] Erdős, P., Some of my favorite problems and results. The mathematics of Paul Erdős, I, Algorithms Combin. 13, Springer (1997), 47--67; the $r_2(3,n)$ bounds and the asymptotic-formula sentence, printed p. 62. Library home: erdos_1997_some_my_favorite_problems_results; paged at problem_p62.
• [HHHKP26] Harris, S., Hefty, Z., Horn, P., King, D. and Pfender, F., A short proof that $R(3,k)=\Theta(k^2/\log k)$. arXiv:2609.01782 (v1 1 September 2026, 7 pages). A lead, not filed in the library: Theorem 1.2 gives $R(3,k)\ge(\frac1{200}+o(1))k^2/\log k$ by a simplified two-bite construction and the note reproduces Shearer's proof of the upper bound.

Formalization. None. formal-conjectures has no file ErdosProblems/165.lean (its directory FormalConjectures/ErdosProblems/ listed 672 entries on 2026-09-18, and no file 165 had been added by 2026-10-07); the site's indicator reads "Formalised statement? No (create one)", and the community database records the problem open, not formalized, with no formal proof and OEIS A000791 (its entry last updated 31 August 2025).

Current assessment

The question (site formulation accessed 2026-09-18). The statement above; labeled OPEN, with the site's note that no finite computation can settle it; prize offered; last edited 7 March 2026. The commentary, restated here, says that for some constant $c>0$ and large $k$, $(c+o(1))k^2/\log k\le R(3,k)\le(1+o(1))k^2/\log k$, with the lower bound due to Kim [Ki95] and the upper bound to Shearer [Sh83], who improved an earlier bound of Ajtai, Komlós and Szemerédi [AKS80]; that Kim's proof gave $c\ge1/162$, that $c\ge1/4$ was proved independently by [BoKe21] and [PGM20], the latter conjecturing that $1/4$ is the truth, and that [CJMS25] and [HHKP25] raised the constant to $1/3$ and then $1/2$, both conjecturing that $c=1/2$ is the right asymptotic; it points to Problem 544, to Problem 986 for the general case and to Problem 1013 for a related function. The thread: comments of 1 and 9 October 2025 reporting the $1/3$ bound and the $1/2$ conjecture, of 23 October 2025 reporting the $1/2$ bound (marked by the site as addressed), of 1 December 2025 on a reference that failed to load, and of 10 June 2026 linking the web page of Trellis, an autoformalization system, as holding a Lean formalization of [HHKP25], recorded below as a lead. The proof-claim tab is empty.

The origins. [Er71] item 6, printed p. 99: after defining $f(l,n)$ as the least order forcing a $K_l$ or $n$ independent points and printing (2) $c_3n^2\log n/\log\log n<f(3,n)<c_4n^2(\log n)^2$ (a misprint: it repeats the bounds that display (1) on p. 98 gives for $g(3,n)$, the least order of a triangle-free graph of chromatic number $n$, whereas the 1961 and 1968 results are $n^2/(\log n)^2\ll f(3,n)\ll n^2\log\log n/\log n$; recorded as printed), "It would be desirable to improve (2) and to obtain an asymptotic formula for $f(l,n)$." [Er78] printed p. 34: "Graver, Yackel and I proved that (4) $c_1n^2/(\log n)^2<r(C_3,K_n)<c_2n^2\log\log n/\log n$. It would be interesting to obtain an asymptotic formula for $r(C_3,K_n)$, but this will probably be very difficult." [Er81c] printed p. 10, item (5): $c_1n^2/(\log n)^2<r(3,n)<c_2n^2/\log n$, "The lower bound is due to me. The upper bound was proved very recently by Ajtai, Komlós and Szemerédi who improved the previous bound $cn^2\log\log n/\log n$ of Graver and Yackel"; and p. 11, after the Erdős--Sós questions of Problem 544: "All these results would easily follow if one could get a good asymptotic formula with a good error term for $r(n,3)$, but needless to say this is nowhere in sight." [Er90b] printed p. 18, in the chapter's notation $F_2(3,k)$ for $R(3,k)$, displays as (13) the bounds $c_1k^2/(\log k)^2<F_2(3,k)<c_2k^2/\log k$, credits the upper bound to Graver and Yackel with an extra $\log\log k$ factor, which Erdős's text places in the denominator (a slip: the Graver--Yackel bound is $ck^2\log\log k/\log k$, with the factor in the numerator, as the [Er78], [Er81c] and [Er93] passages and the 1980 introduction state), and says that Ajtai, Komlós and Szemerédi removed that factor by a new method, which he calls a great breakthrough; after a paragraph on their independence-number theorem he writes "It would be very desirable to get an asymptotic formula for $F(3,k)$", adding that there may be no exact formula at all, just as no useful closed formula is known for the $n$-th prime, with a prize offered for it together with the $R(4,k)$ conjecture (14) of Problem 166. [Er97c] printed p. 62, in its notation $r_2(3,n)$, states the order of magnitude $c_1n^2/\log n<r_2(3,n)<c_2n^2/\log n$ as now known, credits the upper bound to Ajtai, Komlós and Szemerédi and the recent lower bound to Kim's probabilistic argument, and ends "It would be nice to have an asymptotic formula for $r_2(3,n)$", with no prize attached to the wish there. [Er93] printed pp. 338--339 recalls his 1961 probabilistic bound, printed as (7) $r(3,n)<cn^2/(\log n)^2$, his offer of a prize for a proof of $r(3,n)/n^2\to0$ (printed with $\to\infty$), which would have been the first improvement on (6), Szekeres's bound $r(3,n)\le\binom{n+1}2$, Graver and Yackel's $r(3,n)<cn^2\log\log n/\log n$ and the Ajtai--Komlós--Szemerédi bound, printed as (8) $r(3,n)>cn^2/\log n$, calls (7) and (8) the current best bounds, and ends "I would guess that (8) is closer to the truth than (7) but an asymptotic formula for $r(3,n)$ is not in sight and probably will be very difficult", with no prize attached to the wish. The inequality signs of (7) and (8) and the arrow of the prize offer are as printed; (7) is the 1961 lower bound and (8) the 1980 upper bound, so the printed directions of all three are reversed (an observation made here). The site's [Er61] passage, Part II, item 4 (printed pp. 240--241), is quoted under References.

The bounds map, one result page per bound. Lower bounds, in order: Kim's Theorem 1.1 (typescript p. 1): every large $n$ has a triangle-free graph with $\alpha\le9\sqrt{n\log n}$, whence $R(3,t)\ge c(1-o(1))t^2/\log t$ with $c=1/162$ (p. 2, an unlabeled consequence), by the semirandom nibble; Bohman and Keevash's Theorem 1.3 (p. 2), $R(3,t)>(\frac14-o(1))t^2/\log t$, from the terminal graph of the triangle-free process (independence number $(1+o(1))\sqrt{2n\log n}$, Theorem 1.2); Fiz Pontiveros, Griffiths and Morris's Theorem 1.2 (p. 4), $(\frac14-o(1))k^2/\log k\le R(3,k)\le(1\pm o(1))k^2/\log k$, the same process analyzed independently, with their Conjecture 1.3 that $1/4$ is the truth; Campos, Jenssen, Michelen and Sahasrabudhe's Theorem 1.1 (p. 2), $R(3,k)\ge(\frac13+o(1))k^2/\log k$, by a steered triangle-free process run from a blown-up random seed graph, which disproves Conjecture 1.3; and Hefty, Horn, King and Pfender's Theorem 1.2 (p. 2), $R(3,k)\ge(\frac12+o(1))k^2/\log k$, from a random overlay of two blow-ups of a random graph with no nibble (their Theorem 1.3: triangle-free graphs on $n$ vertices with $\alpha<(1+\varepsilon)\sqrt{n\log n}$). Upper bound: Shearer's $R(3,k)\le(1+o(1))k^2/\log k$ [Sh83], from his Theorem 1 (printed p. 83): a triangle-free graph on $n$ vertices with average degree $d$ has $\alpha\ge nf(d)$, $f(d)=(d\ln d-d+1)/(d-1)^2$, which is the $\alpha(G)\ge(1-o(1))n\log d/d$ of [PGM20] display (2); a triangle-free graph with $\alpha\le k-1$ has every degree at most $k-1$, so $n\le(k-1)/f(k-1)=(1+o(1))k^2/\log k$, an elementary step recorded on the result page (the paper prints no Ramsey number). It sharpens the Ajtai--Komlós--Szemerédi bound, Theorem 2 of [AKS80] (printed p. 355): a triangle-free graph with $n$ vertices and average degree $t$ has $\alpha(G)\ge0.01(n/t)\ln t$ (stated on p. 83 of [Sh83] as $\alpha>n\ln d/(100d)$ for $d\ge d_0$, credited there to the same authors' Sidon-sequence paper), from which their Theorem 3 (printed p. 358), $R(3,x)<100x^2/\ln x$, follows by the same degree step, which the paper prints in four sentences. The 1980 introduction (p. 354) places it against "$cx^2/(\ln x)^2<R(3,x)<cx^2\ln\ln x/\ln x$", Erdős's 1961 lower bound and Graver and Yackel's 1968 upper bound, so the theorem removed the $\log\log$ factor. Graver and Yackel's paper is recorded at statement depth: Proposition 9 (printed p. 154), $R(3,y)\le By^2\log\log y/\log y$ in the paper's convention, in which $R(3,y)$ is the largest order of a triangle-free graph with no $y$ independent vertices, one less than the usual Ramsey number; its proof (p. 156) has been checked except for the estimates of Lemma 9's proof (p. 155), and the bracket of its display (5) gives the constant $2+o(1)$, which the paper does not state. The same bound is quoted in the introductions of [CJMS25] (display (1) and Section 1.1), [PGM20] (Theorem 1.2), [BoKe21] (p. 2), [HHKP25] (p. 1) and [Ki95] (display (1)); Erdős's 1961 paper has a library card (erdos_1961_graph_theory_probability), and its lower bound is quoted here from the same introductions. The conjectured formula is Conjecture 1.2 of [CJMS25], $R(3,k)=(\frac12+o(1))k^2/\log k$, restated as Conjecture 1.1 of [HHKP25], whose Theorem 1.2 is its lower half; its upper half would follow, [HHKP25] p. 2 says, from a conjecture of Davies, Jenssen, Perkins and Roberts relating the maximum and average sizes of independent sets in triangle-free graphs. Acceptance: [Ki95], [BoKe21] and [PGM20] are refereed (per their journal records; the page numbers cited are those of a typescript and two arXiv versions, which have not been compared with the journal texts); [CJMS25] and [HHKP25] are preprints with no journal record, so the constant $1/2$ carries the preprint qualification, while the refereed $1/4$ stands without it. Morris's 2026 survey [Mo26], Theorem 1.2 (p. 2), states the two bounds together, $(\frac12+o(1))k^2/\log k\le R(3,k)\le(1+o(1))k^2/\log k$ as $k\to\infty$, with "the upper bound in Theorem 1.2 was proved by Shearer [86] in 1983, and the lower bound very recently by Hefty, Horn, King and Pfender [59]", the bounds now differing "by only a factor of $2+o(1)$"; a published plenary lecture's attestation of the preprint's bound, not an independent review, so the qualification stands. Depth: every statement named here is checked against its source; of the proofs, only the one-page proof of Shearer's Theorem 1 and the four-sentence proof of Theorem 3 of [AKS80] are checked.

Leads (with provenance, not status). (1) The preprint [HHHKP26] (arXiv:2609.01782v1, 1 September 2026), not filed in the library: its Theorem 1.2 proves only the constant $1/200$ by a shorter version of the two-bite construction and its Theorem 1.3 gives $\alpha\le(10+o(1))\sqrt{n\log n}$; it changes no bound and is a candidate for filing if it is refereed. (2) The thread comment of 10 June 2026 links the web page of Trellis, an autoformalization system built on LLM agents, which lists a Lean formalization of [HHKP25] produced during the system's early development and finished by hand; nothing from it was built or checked in this corpus, the community database records no formal proof, and the bound it formalizes, Theorem 1.2's constant $1/2$, does not answer the question, which asks for an asymptotic formula, so it gets no claim page. (3) [CJMS25] p. 4 announces a companion paper with a construction from the sum-free process on $\mathbb F_2^d$ that would conjecturally give the constant $1/2$ along an infinite sequence of $k$; not found in the search below. (4) Semantic Scholar lists 20 records citing [HHKP25] and 24 citing [CJMS25] (by title,): the 2026 items concern Erdős--Rogers functions, hypergraph Ramsey numbers, cycle-complete numbers, the odd Hadwiger conjecture and Morris's survey; none claims a constant above $1/2$ or a matching upper bound.

Search scope. None of the routes below found an asymptotic formula, a lower constant above $1/2$, an upper constant below $1$, a journal version of the two 2025 preprints, or a proof claim.

• The site: problem page, discussion thread and proof-claim tab; the formal-conjectures directory (no file); the community database; OEIS A000791.
• arXiv: the abstract pages of 2505.13371 (one version), 2510.19718 (three versions), 1302.6279 (two versions), 2609.01782 (one version) and the API record of 1302.5963 (two versions; its abstract page answered HTTP 406); the API queries abs:"R(3,k)" AND abs:Ramsey sorted by date (ten records, the newest Morris's 2026 survey) and abs:"off-diagonal Ramsey" (20 records; nothing newer than the two 2025 preprints on $R(3,k)$).
• Crossref: the records of [Ki95], [BoKe21], [PGM20], [Sh83] and [AKS80]; bibliographic queries for the titles of [CJMS25] and [HHKP25] (no journal record).
• Semantic Scholar: the citation lists of [HHKP25] (20 records) and [CJMS25] (24 records); the endpoint answered HTTP 429 to a further query and was not retried.
• One request each to the publisher's full-text links of [Sh83] and [AKS80] (both HTTP 403).
• The primary sources at the pages cited: [HHKP25] pp. 1--3, [CJMS25] pp. 1--4, [BoKe21] pp. 1--3, [PGM20] pp. 1--5, [Ki95] typescript pp. 1--3, [Er71] pp. 98--99, [Er78] p. 34, [Er81c] pp. 10--11, [Er90b] p. 18, [Mo26] p. 2 and [HHHKP26] pp. 1--2.

Not searched: MathSciNet, zbMATH, Google Scholar, X. [Er97c], [Er93], [Sh83], [AKS80] and [GrYa68] were filed in the library after the search. Erdős's 1961 paper for the lower bound has a library card (graph_coloring/erdos_1961_graph_theory_probability) and is not cited directly; [Er61] is the problem collection, whose passage is quoted under References.

Remaining gaps. (1) The constant is open between $1/2$ and $1$; the lower half rests on two unrefereed preprints and the upper half on Shearer's 1983 paper, recorded at statement depth with its proof checked; the reopening condition is a proof of either half of Conjecture 1.2 or a refereed version of [HHKP25]. (2) Proof coverage is statements only for the five lower-bound results; the one-page proof of Shearer's Theorem 1 has been checked, but nothing is compiled or reviewed. (3) The [Er61] passage (Part II, item 4, printed pp. 240--241) is quoted under References, and the [Er71], [Er78], [Er81c], [Er90b], [Er93] and [Er97c] passages above, each by printed page. (4) The methods (nibble, triangle-free process, seeded process, two bites) are named with pointers only.

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.

• erdos_1971_unsolved_problems_graph_theory_combinatorial_analysis
• erdos_1978_problems_results_combinatorial_analysis_combinatorial_number
• erdos_1993_my_favorite_solved_unsolved_problems_graph_theory
• erdos_1961_graph_theory_probability
• graver_yackel_1968_graph_theoretic_results_associated_ramsey_theorem
• graver_yackel_1968_graph_theoretic_results_associated_ramsey_theorem / proposition_9
• erdos_1961_unsolved_problems
• ajtai_1980_note_ramsey_numbers
• ajtai_1980_note_ramsey_numbers / theorem_2
• ajtai_1980_note_ramsey_numbers / theorem_3
• bohman_2021_dynamic_concentration_triangle_free_process
• bohman_2021_dynamic_concentration_triangle_free_process / theorem_1_2
• bohman_2021_dynamic_concentration_triangle_free_process / theorem_1_3
• campos_2025_new_lower_bound_ramsey_numbers
• campos_2025_new_lower_bound_ramsey_numbers / conjecture_1_2
• campos_2025_new_lower_bound_ramsey_numbers / theorem_1_1
• erdos_1981_new_problems_results_graph_theory_other
• erdos_1990_problems_results_graphs_hypergraphs_similarities_differences
• erdos_1997_some_my_favorite_problems_results
• erdos_1997_some_my_favorite_problems_results / problem_p62
• fizpontiveros_2020_triangle_free_process_ramsey_number
• fizpontiveros_2020_triangle_free_process_ramsey_number / conjecture_1_3
• fizpontiveros_2020_triangle_free_process_ramsey_number / theorem_1_2
• fizpontiveros_2020_triangle_free_process_ramsey_number / theorem_2_12
• hefty_2025_improving_just_two_bites
• hefty_2025_improving_just_two_bites / theorem_1_2
• hefty_2025_improving_just_two_bites / theorem_1_3
• kim_1995_ramsey_number_has_order_magnitude
• kim_1995_ramsey_number_has_order_magnitude / theorem_1_1
• morris_2026_recent_results_ramsey_theory
• shearer_1983_note_independence_number_triangle_free_graphs
• shearer_1983_note_independence_number_triangle_free_graphs / theorem_1