Problem 1011

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claims/: The 5 claim pages of Problem 1011, one per claimant's result; the problem's standing derives from them.

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Statement. Let $f_r(n)$ be minimal such that every graph on $n$ vertices with $\geq f_r(n)$ edges and chromatic number $\geq r$ contains a triangle. Determine $f_r(n)$.

Formulation. The site's wording as accessed 2026-09-18 (page last edited 6 December 2025). Three conventions are fixed on this page, each an observation made on it. First, for triangle-free graphs the condition "chromatic number $\ge3$" is the same as "not bipartite", since a graph is bipartite exactly when it has a proper $2$-coloring; Erdős's 1962 paper states its lemma for graphs that are "not even" (not bipartite) and never mentions the chromatic number. Second, when some triangle-free graph on $n$ vertices has chromatic number at least $r$, $f_r(n)$ is one more than the largest number of edges of such a graph (every graph with more edges and chromatic number at least $r$ has a triangle, and the extremal graph shows that one edge fewer does not suffice); the sources state their results in the maximum-edge form, and this page converts them by adding one. Third, when no triangle-free graph on $n$ vertices has chromatic number at least $r$, the condition is vacuous and the literal minimum is $0$; this happens for $r=4$ and $n\le10$, the Grötzsch graph on $11$ vertices being the smallest triangle-free $4$-chromatic graph. Erdős's 1971 list writes the same function as $u_r$ with the same indexing ($u_2$ is Turán's $[\tfrac14n^2]+1$ and $u_3=[\tfrac14(n-1)^2]+2$); the site's $f_r$ and the paper's $u_r$ agree. The site's $g(r)$ is Simonovits's $\hat g_3(t)$ at $t=r$, defined in Theorem 2.7 of [Si74] (p. 358) as "the largest integer $m$ such that for any graph $G$ not containing $K_3$ and having chromatic number $\ge t$, at least $m$ vertices of $G$ must be omitted to get a 2-chromatic graph"; the site's own description of $g(r)$, the largest $m$ such that every triangle-free graph of chromatic number at least $r$ needs at least $m$ vertex deletions to become bipartite (the minimum, over such graphs, of the fewest deletions that make the graph bipartite), says the same.

Status. Open. The exact values known are $f_2(n)=\lfloor n^2/4\rfloor+1$ (Turán's theorem, as the site says and as [Er62d] p. 122 records), $f_3(n)=\lfloor(n-1)^2/4\rfloor+2$ for $n\ge5$ (the upper bound is Lemma 1 of [Er62d], the Erdős--Gallai and Andrásfai theorem, and the matching triangle-free non-bipartite graph with one edge fewer is on p. 124 of the same paper and is the graph $H_0$ of [RWWY24]), and $f_4(n)=\lfloor(n-3)^2/4\rfloor+6$ for $n\ge90$ (Theorem 1.4 of [RWWY24], arXiv v2 of 19 October 2025, a preprint; the blow-ups of the Grötzsch graph give the matching lower bound). The site prints the range $n\ge150$ for the last value; v2 prints $n\ge90$, a site-versus-source discrepancy recorded below. For general $r$, Theorem 2.7 of [Si74] (p. 358; Simonovits attributes it to his thesis and prints no proof) gives $f_r(n)=\tfrac{n^2}4-\tfrac{g(r)}2n+O(1)$ with $g(r)=\hat g_3(r)$, and Remark 2.8(a) of the same paper (p. 359) asserts without proof that $\tfrac{\log r}{\log\log r}r^2\ll g(r)\ll(\log r)^2r^2$; the site reports both, and, from its discussion thread, that $g(r)\asymp r^2\log r$; the inputs to the thread's derivation, Theorem 1 of [DaIl22] and Theorem 1.3 of [HHKP25], are checked on this page, but the derivation itself is a forum argument recorded with provenance and unverified. The three literature determinations have claim pages: Turán for $f_2(n)$ (accepted, a refereed partial claim), Erdős and Gallai for $f_3(n)$ (accepted, a refereed partial claim) and Ren, Wang, Wang and Yang for $f_4(n)$ with $n\ge90$ (a pending partial claim; the paper is a preprint). No source found determines $f_r(n)$ for any $r\ge5$, or $f_4(n)$ for $n<90$, beyond forum claims of September 2026 for $r=4$ and for $r=5$ with $n\ge80$, recorded on two further pending partial claim pages below. No proof or disproof of a general formula was found in the search whose scope the Current assessment records; this is a bounded negative finding, not a certificate of openness.

Source. erdosproblems.com/1011, accessed 2026-09-18: the problem page (labeled OPEN, the site's label for a problem that is open and not settled by a finite computation; last edited 6 December 2025; source key [Er71], with [Er62d], [Si74], [DaIl22], [HHKP25] and [RWWY24] cited in the commentary; the page thanks three contributors by name), its discussion thread (ten comments from 13 October 2025 to 19 September 2026) and its empty proof-claim tab. Cite as: T. F. Bloom, Erdős Problem #1011, https://www.erdosproblems.com/1011, accessed 2026-09-18.

References.

• [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 3, printed p. 98 (PDF p. 2 of the Rényi archive's scan). Library home: erdos_1971_unsolved_problems_graph_theory_combinatorial_analysis; the passage is paged at item_3.
• [Er62d] Erdős, P., On a theorem of Rademacher-Turán. Illinois J. Math. 6 (1962), no. 1, 122--127, doi:10.1215/ijm/1255631811 (Crossref record). Lemma 1, p. 123; the extremal example, p. 124. Library home: erdos_1962_theorem_rademacher_turan (the Rényi archive's scan 1962-09.pdf); paged at lemma_1.
• [RWWY24] Ren, S., Wang, J., Wang, S. and Yang, W., Extremal triangle-free graphs with chromatic number at least four. arXiv:2404.07486 (v1 11 April 2024; v2 19 October 2025, "the proof is slightly improved", 14 pages). A preprint. Theorems 1.2 and 1.4, pp. 1--2. Library home: ren_2024_extremal_triangle_free_graphs_chromatic_number; paged at theorem_1_4 and theorem_1_2.
• [Si74] Simonovits, M., Extremal graph problems with symmetrical extremal graphs. Additional chromatic conditions. Discrete Math. 7 (1974), no. 3--4, 349--376, doi:10.1016/0012-365X(74)90044-2 (received 12 September 1973, original version 30 March 1972; Crossref record, carrying the publisher's open-archive license dated 2013-07-17). Theorem 2.7, p. 358; Remark 2.8, p. 359; the site cites "the discussion on p. 358". Library home: simonovits_1974_extremal_graph_problems_symmetrical_extremal_graphs (the publisher's open-archive copy; the card carries the row for this problem); paged at theorem_2_7 and remark_2_8. The site's reference text misspells the title's first word ("Extermal").
• [DaIl22] Davies, E. and Illingworth, F., The $\chi$-Ramsey problem for triangle-free graphs. SIAM J. Discrete Math. 36 (2022), no. 2, 1124--1134, doi:10.1137/21M1437573 (published online 28 April 2022; Crossref record); arXiv:2107.12288v2 (28 January 2022, 13 pages). Theorem 1, p. 3 of the preprint. Library home: davies_2022_ramsey_problem_triangle_free_graphs; paged at theorem_1.
• [HHKP25] Hefty, Z., Horn, P., King, D. and Pfender, F., Improving $R(3,k)$ in just two bites. arXiv:2510.19718 (v3 19 February 2026, 18 pages); a preprint. Theorem 1.3, p. 2. Library home: hefty_2025_improving_just_two_bites; paged at theorem_1_3.

Formalization. None in formal-conjectures or in Alexeev's repository: no file ErdosProblems/1011.lean exists in google-deepmind/formal-conjectures (the directory was listed on 2026-09-18 and the path was absent on main on 2026-10-07); the repository's issue 1069 ("Erdős Problem 1011", opened 14 October 2025) asks for a statement, and the claimant of the forum claims below commented on it on 10 and 19 September 2026 with a proposed statement and the announcements of the proofs. The site's indicator reads "Formalised statement? No", and the community database (read 2026-09-18 and 2026-10-06) lists the problem as open as of its last update, 10 September 2025, unformalized, with no formal proof. Forum comments of 10 and 19 September 2026 announce an external Lean development for the cases $r=4$ (every $n$) and $r=5$ ($n\ge80$); each has its own claim page, 2026_09_10_kentakitamura and 2026_09_19_kentakitamura, both claimed and partial; the repository is neither built nor kernel-checked in this corpus.

Current assessment

The question (site formulation, accessed 2026-09-18). The statement above; OPEN; last edited 6 December 2025. The site's commentary, in this page's words: Turán's theorem gives $f_2(n)=\lfloor n^2/4\rfloor+1$; Erdős and Gallai [Er62d] determined $f_3(n)=\lfloor\frac14(n-1)^2\rfloor+2$; Simonovits's thesis (the site points to the discussion on p. 358 of [Si74]) gives $f_r(n)=\frac{n^2}4-\frac{g(r)}2n+O(1)$, where $g(r)$ is the largest $m$ such that every triangle-free graph of chromatic number at least $r$ needs at least $m$ vertex deletions to become bipartite; [Si74] records $\frac{\log r}{\log\log r}r^2\ll g(r)\ll(\log r)^2r^2$; a thread observation, adopted by the site, derives $g(r)\asymp r^2\log r$ from other results, more precisely $(1/2-o(1))r^2\log r\le g(r)\le(2+o(1))r^2\log r$, the lower bound from Davies and Illingworth [DaIl22] (the site refers to Problem 1104) and the upper bound from the $R(3,k)$ constructions of Hefty, Horn, King and Pfender [HHKP25]; and Ren, Wang, Wang and Yang [RWWY24] determined $f_4(n)=\lfloor\frac{(n-3)^2}4\rfloor+6$ for $n\ge150$. The thread and the proof-claim tab are described below; the community database record says open.

The origin. Item 3 of the 1971 list (p. 98), after Erdős's edge-disjoint-triangles theorem (Problem 1009), reads: "In view of my theorem with Gallai the following question could be asked: what is the smallest integer $u_r$ so that every $G(n;u_r)$ which has chromatic number $\geqslant r$, contains a triangle? $u_2=[\tfrac14n^2]+1$ (this is the well known theorem of Turán) and $u_3=[\tfrac14(n-1)^2]+2$. $u_4$ is unknown [5].†", with the note added in proof "† Simonovits determined $u_r$." The footnote is Erdős's printed attestation of the result the site takes from [Si74]; the theorem it refers to is Theorem 2.7 of [Si74] (p. 358), which Simonovits attributes to his thesis.

The case $r=3$ (refereed). Lemma 1 of [Er62d] (p. 123): "Every $G^{(n)}_{f(n-1)+2}$ which is not even contains a triangle", where $f(n-1)=\lfloor(n-1)^2/4\rfloor$ and "even" means every circuit has an even number of edges; "Lemma 1 was found jointly by Gallai and myself. (The lemma was also found by Mr. Andrásfai independently.)" The proof (pp. 123--124) bounds the edges of a non-even triangle-free graph by $2k+1+2(n-2k-1)+f(n-2k-1)\le f(n-1)+1$, where $2k+1$ is the length of a shortest odd circuit, "equality only for $2k+1=5$", so the lemma holds for every edge count at least $f(n-1)+2$ and $f_3(n)\le\lfloor(n-1)^2/4\rfloor+2$. Sharpness: p. 124 gives, for a graph whose shortest odd circuit has $2k+1$ vertices, $k>1$, the bound $2n-2k-1+f(n-2k-1)$ and "the following simple example shows that this result is best possible" (vertices $\alpha_1,\dots,\alpha_v$, $\beta_1,\dots,\beta_u$, $\gamma_1,\dots,\gamma_{2k+1}$ with $v=[(n-2k-1)/2]$ and $u=n-2k-1-v$; the paper prints "$u=n-[(n-2k-1)/2]$", a misprint, since that value gives $v+u+2k+1=n+2k+1$ vertices, while $u=n-2k-1-v$ gives $n$ vertices and exactly $vu+2(v+u)+2k+1 =f(n-2k-1)+2(n-2k-1)+2k+1=2n-2k-1+f(n-2k-1)$ edges, the stated bound; the edges are all $\alpha_i\beta_j$, $\gamma_1$ and $\gamma_3$ to every $\alpha_i$, $\gamma_2$ and $\gamma_4$ to every $\beta_j$, and the odd circuit on the $\gamma$'s); at $k=2$ the bound is $2n-5+f(n-5)=f(n-1)+1$ (a two-line check made on this page: $f(m)-f(m-1)=[m/2]$, so $f(n-1)-f(n-5)=2n-6$ for every $n$), so the example is a triangle-free non-bipartite graph with $\lfloor(n-1)^2/4\rfloor+1$ edges for every $n\ge5$. The same value is Theorem 1.2 of [RWWY24] with its graph $H_0$ (pp. 1--2), stated in the maximum-edge form. Hence $f_3(n)=\lfloor(n-1)^2/4\rfloor+2$ for $n\ge5$; for $n\le4$ no triangle-free graph is non-bipartite, so the condition is vacuous. The determination is an accepted partial claim on its page, 1962_03_01_erdos (refereed; the site's commentary credits it, which on a problem the site labels OPEN is not acceptance evidence). Turán's theorem, the case $r=2$, is an accepted partial claim on its page, 1941_01_01_turan.

The case $r=4$ (a preprint). Theorem 1.4 of [RWWY24] (p. 2 of v2): "Let $G$ be a graph on $n$ vertices with $n\ge90$. If $G$ is triangle-free and $\chi(G)\ge4$, then $e(G)\le\lfloor\frac{(n-3)^2}4\rfloor+5$, with equality if and only if $G\in\mathcal G(n)$ up to isomorphism", where $\mathcal G(n)$ is the family of blow-ups of the Grötzsch graph with $\sum|V_i|=\lfloor\frac{n-3}2\rfloor$ (or $\lceil\frac{n-3}2\rceil$) and $|W|=\lceil\frac{n-7}2\rceil$ (or $\lfloor\frac{n-7}2\rfloor$), "each graph in $\mathcal G(n)$ is a triangle-free $4$-chromatic graph with $\lfloor\frac{(n-3)^2}4\rfloor+5$ edges". Conversion (made on this page): every graph on $n\ge90$ vertices with $\chi\ge4$ and at least $\lfloor(n-3)^2/4\rfloor+6$ edges contains a triangle, and the graphs of $\mathcal G(n)$ show that one edge fewer does not force one; so $f_4(n)=\lfloor(n-3)^2/4\rfloor+6$ for $n\ge90$, the site's formula. The v2 text prints $n\ge90$ where the site prints $n\ge150$; the thread comment of 13 October 2025 quotes $150$ from the preprint as it then stood, and v2 (19 October 2025) says "the proof is slightly improved", which is consistent with the site's figure having come from v1, which this corpus has not consulted. The paper is a preprint (no journal record found), so the value carries the preprint qualification; the proof (Section 4, through a vertex-stability form of Mantel's theorem, Theorem 1.5, which Section 3 proves from the lemmas of Section 2) is not examined in this corpus. The determination is a pending partial claim on its page, 2024_04_11_ren_wang_wang_yang.

General $r$ (at statement depth). Theorem 2.7 of [Si74] (p. 358), introduced by Erdős's "Problem. What is the maximum number of edges, a graph of $n$ vertices and chromatic number $\ge t$ can have if it does not contain $K_3$?" and by "I showed [13] that": "Let $f_t(n;K_3)$ denote the maximum in the problem above. Then $f_t(n;K_3)=\frac14n^2-\hat g_3(t)\frac12n+O(1)$, where $\hat g_3(t)$ is the largest integer $m$ such that for any graph $G$ not containing $K_3$ and having chromatic number $\ge t$, at least $m$ vertices of $G$ must be omitted to get a 2-chromatic graph." With $t=r$ and $f_r(n)=f_r(n;K_3)+1$ (the second convention above) this is the site's expansion $f_r(n)=\tfrac{n^2}4-\tfrac{g(r)}2n+O(1)$ with $g(r)=\hat g_3(r)$. The paper's [13] is Simonovits's thesis, "On the structure of extremal graphs, Ph.D. Thesis, Library of Acad. Sci. Hungar. (in Hungarian)", not held; the paper prints no proof of Theorem 2.7. Remark 2.8(a) (p. 359) asserts, "Comparing $\hat g_3$ and $g_3$ of [1], one can easily prove that $c_1t^2\log t/\log\log t<\hat g_3(t)<c_2t^2(\log t)^2$", the site's bounds $\tfrac{\log r}{\log\log r}r^2\ll g(r)\ll(\log r)^2r^2$; [1] is Erdős's 1959 paper Graph theory and probability, and no proof is printed. Remark 2.8(c) calls the $K_4$ analogue "an essentially more difficult problem the exact solution of which is unknown to me". The thread comment of 13 October 2025 named "Theorem 2.7" of [Si74] with "an explicit function $g_3(t)$" and the thesis, On the structure of extremal graphs, as its source, and said its references were located by GPT-5; the theorem number and the thesis title are as printed, and the function is printed with a hat, $\hat g_3$, which the paper distinguishes from the $g_3$ of [1]. The thread's derivation of $g(r)\asymp r^2\log r$ (the accounts zach hunter, Wouter CvB and the site's maintainer, 25--26 October 2025): $g(r)\le\min\{R(3,k)-2k:R(3,k)/k>r-1\}$ gives the upper bound from the triangle-free graphs with small independence number of [HHKP25], and $g(r)\ge\min{n:\text{some triangle-free graph on }n\text{ vertices has chromatic number }r-2}$ gives the lower bound from [DaIl22]; after two corrections of constants within the thread the site prints $(1/2-o(1))r^2\log r\le g(r)\le(2+o(1))r^2\log r$. What is checked on this page is only the two inputs: Theorem 1 of [DaIl22] (p. 3 of the preprint; SIAM J. Discrete Math. 2022, refereed): "As $n\to\infty$, any triangle-free graph on $n$ vertices has chromatic number at most $(2+o(1))\sqrt{n/\log n}$", so no triangle-free graph on $n$ vertices has chromatic number $r$ once $r>(2+o(1))\sqrt{n/\log n}$, and $f_r(n)$ is vacuous beyond that range; and Theorem 1.3 of [HHKP25] (p. 2; a preprint): "For all $\varepsilon>0$, there exists an $n_0=n_0(\varepsilon)$ so that for $n\ge n_0$ there exists a triangle-free graph $G$ on $n$ vertices with independence number $\alpha(G)<(1+\varepsilon)\sqrt{n\log n}$", whose chromatic number is at least $n/\alpha(G)>\sqrt{n/\log n}/(1+\varepsilon)$ (since $\alpha(G)\chi(G)\ge n$), so triangle-free graphs of chromatic number of order $\sqrt{n/\log n}$ exist and $f_r(n)$ is a nontrivial question up to $r$ of that order. Neither theorem states anything about $f_r(n)$ itself; their bearing on this problem is through the thread's reduction, which is not verified in this corpus. The exact value of $f_r(n)$ is not known for any $r\ge5$ from any source found. Neither Simonovits's expansion nor the thread's bounds on $g(r)$ determines $f_r(n)$ for any $r$, so neither is a claim.

Forum claims (pending partial claims, not status). A thread comment (08:26 on 10 September 2026, the account KentaKitamura, signed Kenta Kitamura) announces a Lean 4 development said to determine $f_4(n)$ for every $n$: $f_4(n)=0$ for $n\le10$, $f_4(11)=21$ and $f_4(n)=\lfloor(n-3)^2/4\rfloor+6$ for $n\ge12$, extending the preprint's range, with a formal-conjectures proposal (a comment on the issue 1069 above) and a public repository, at the commit the claim page pins, neither built nor kernel-checked in this corpus. The claim page 2026_09_10_kentakitamura records what the repository contains, the comment's disclosure of assistance from OpenAI Codex and ChatGPT Astra, and the standing: unrefereed, not accepted by the site (whose page prints $n\ge150$) or by the community database, and covering the case $r=4$ alone, so the problem's standing is unaffected. A further comment of the same account (11:40 on 19 September 2026) announces, in the same repository, a kernel-checked Lean proof of $f_5(n)=\lfloor n^2/4\rfloor-3n+15$ for every $n\ge80$, with the same disclosure, and says that it leaves $r=5$ with $n<80$ and the general problem open; its claim page 2026_09_19_kentakitamura records the standalone Lean file the comment pins, at that commit, neither built nor kernel-checked in this corpus, and the same standing: unrefereed, accepted by nobody, and covering one case of one $r$.

Search scope. None of the routes below found a determination of $f_r(n)$ for any $r\ge5$ or a journal version of [RWWY24]; the publisher's open-archive copy of [Si74] was accessed.

• The site: problem page, discussion thread and proof-claim tab; the formal-conjectures directory listing and recursive tree at the pinned commit (no file 1011) and its issue 1069 through the GitHub API; the community database at the pinned commit; the repository named in the thread, at its head commit, through the GitHub API.
• arXiv API: the records of 2404.07486 (v1 11 April 2024, v2 19 October 2025; no journal reference), 2107.12288 (v2, DOI 10.1137/21M1437573) and 2510.19718 (v3 19 February 2026); the search abs:"triangle-free" AND abs:"chromatic number" AND (abs:extremal OR abs:"number of edges") (14 records, read by title; none determines $f_r(n)$).
• Crossref: the records of [Si74] and [DaIl22]; a bibliographic query for [RWWY24]'s title (no journal record; the top hit is a 2027 Discrete Mathematics paper on spectral extremal results for triangle-free graphs with chromatic number at least four, a lead by title only).
• The publisher's open-archive copy of [Si74].
• The primary sources: [Er62d] pp. 122--124, [Er71] p. 98, [RWWY24] pp. 1--3, [DaIl22] pp. 1--3, [HHKP25] p. 2.

Not searched: MathSciNet, zbMATH, Google Scholar, X. Not held: Simonovits's thesis, the journal text of [DaIl22].

Remaining gaps. (1) [Si74] (publisher's open-archive copy) is paged at pp. 358--359, so the expansion in $g(r)$ and its bounds rest on Theorem 2.7 and Remark 2.8(a) at statement depth; the paper prints no proof of either, attributing the theorem to Simonovits's thesis (in Hungarian, not held) and calling the bounds easy, so the thesis remains the only route to a proof of the expansion. (2) The thread's $g(r)\asymp r^2\log r$ is a forum derivation whose inputs are checked and whose reduction is not. (3) [RWWY24] is a preprint; the site's $n\ge150$ against the paper's $n\ge90$ is recorded and not resolved with the site. (4) The Lean claims of 10 and 19 September 2026 are unreviewed and recorded as pending partial claims; if correct they would settle $f_4(n)$ for all $n$ and $f_5(n)$ for $n\ge80$. (5) Proof coverage is statements only, apart from the outline of Lemma 1's proof above. The list of linked library material below is derived from the library links and records no progress.

Known results

• Turán's theorem: $f_2(n)=\lfloor n^2/4\rfloor+1$ (the site; [Er62d] p. 122; an accepted partial claim, claim page).
• Erdős--Gallai (Andrásfai), Lemma 1 (1962) with the p. 124 example: $f_3(n)=\lfloor(n-1)^2/4\rfloor+2$ for $n\ge5$ (an accepted partial claim, claim page); restated as Theorem 1.2 of [RWWY24].
• Erdős 1971, item 3: the question for $u_r$, the values $u_2$, $u_3$, "$u_4$ is unknown", and the note "Simonovits determined $u_r$".
• Simonovits, Theorem 2.7 (1974, attributed to his thesis, no proof printed): $f_r(n)=\tfrac{n^2}4-\tfrac{g(r)}2n+O(1)$ with $g(r)=\hat g_3(r)$, and Remark 2.8(a): $\tfrac{\log r}{\log\log r}r^2\ll g(r)\ll(\log r)^2r^2$, asserted without proof.
• Ren--Wang--Wang--Yang, Theorem 1.4 (preprint): $f_4(n)=\lfloor(n-3)^2/4\rfloor+6$ for $n\ge90$ (a pending partial claim, claim page).
• Kitamura's Lean claims of September 2026 (pending partial claims, r equal to 4 and r equal to 5): $f_4(n)$ for every $n$ and $f_5(n)=\lfloor n^2/4\rfloor-3n+15$ for $n\ge80$; neither built nor kernel-checked in this corpus.
• Davies--Illingworth, Theorem 1 (2022) and Hefty--Horn--King--Pfender, Theorem 1.3 (preprint): the range of $r$ for which $f_r(n)$ is a nontrivial question is of order $\sqrt{n/\log n}$; the thread's $g(r)\asymp r^2\log r$ rests on them. See Problem 1104 for the chromatic number of triangle-free graphs itself.

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_1962_theorem_rademacher_turan
• erdos_1962_theorem_rademacher_turan / lemma_1
• erdos_1971_unsolved_problems_graph_theory_combinatorial_analysis
• erdos_1971_unsolved_problems_graph_theory_combinatorial_analysis / item_3
• ren_2024_extremal_triangle_free_graphs_chromatic_number
• ren_2024_extremal_triangle_free_graphs_chromatic_number / theorem_1_2
• ren_2024_extremal_triangle_free_graphs_chromatic_number / theorem_1_4
• ren_2024_extremal_triangle_free_graphs_chromatic_number / theorem_1_5
• simonovits_1974_extremal_graph_problems_symmetrical_extremal_graphs
• simonovits_1974_extremal_graph_problems_symmetrical_extremal_graphs / remark_2_8
• simonovits_1974_extremal_graph_problems_symmetrical_extremal_graphs / theorem_1
• simonovits_1974_extremal_graph_problems_symmetrical_extremal_graphs / theorem_1_a
• simonovits_1974_extremal_graph_problems_symmetrical_extremal_graphs / theorem_2
• simonovits_1974_extremal_graph_problems_symmetrical_extremal_graphs / theorem_2_2
• simonovits_1974_extremal_graph_problems_symmetrical_extremal_graphs / theorem_2_7
• simonovits_1974_extremal_graph_problems_symmetrical_extremal_graphs / theorem_3
• davies_2022_ramsey_problem_triangle_free_graphs
• davies_2022_ramsey_problem_triangle_free_graphs / theorem_1
• hefty_2025_improving_just_two_bites
• hefty_2025_improving_just_two_bites / theorem_1_3