Problem 1033

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claims/: The 1 claim page of Problem 1033, one per claimant's result; the problem's standing derives from them.

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Statement. Let $h(n)$ be such that every graph on $n$ vertices with $>n^2/4$ many edges contains a triangle whose vertices have degrees summing to at least $h(n)$. Estimate $h(n)$. In particular, is it true that

$$h(n)\geq (2(\sqrt{3}-1)-o(1))n?$$

Formulation. The site's wording(page last edited 3 April 2026, when the former content of Problem 904 moved here). $h(n)$ is the largest value that works, the minimum over all graphs $G$ with $n$ vertices and more than $n^2/4$ edges of the largest degree sum of a triangle of $G$; every such graph has a triangle by Mantel's theorem. In the sources' notation it is Fan's $f(n,e)$ at $e=\lfloor n^2/4\rfloor+1$ (the function is nondecreasing in the edge count, so the minimum is attained at the fewest edges) and Bollobás and Nikiforov's $\Delta_3(n,m)$ at $m=\lfloor n^2/4\rfloor+1=t_2(n)+1$. The page-level question is the estimate; the "in particular" question is yes or no, and its constant $2(\sqrt3-1)\approx1.4641$ is the value of the construction below, so it asks whether that construction is asymptotically best. The thread carries an unreviewed claim that the answer is no (recorded below and on its own claim page as a pending partial claim).

Status. Open. The bounds in hand are $\frac{21}{16}n<h(n)\le2(\sqrt3-1)n+O(1)$, with $21/16=1.3125$ and $2(\sqrt3-1)\approx1.4641$. The lower bound is Fan's Theorem 1 with its Corollary 1.1 (J. Graph Theory 12 (1988), refereed): every graph with $n$ vertices and $e>n^2/4$ edges has a triangle with degree sum $>21e/4n$, so $f(n,\lfloor n^2/4\rfloor+1)>21n/16$ as printed on p. 251 (theorem_1); the earlier lower bound $h(n)\ge(1+\eta)n$ for a fixed $\eta>0$ and large $n$ is Theorem 2 of the Erdős--Laskar note (theorem_2). The upper bound is the site's construction, recomputed below as an authored check; the site attributes it to Erdős and Laskar and says the bound is not explicit in their 1985 note, whose six pages indeed contain no such construction, while Fan's § 2 (pp. 251--252) prints the construction in full with the bound $f(n,e)<4\sqrt{3e}-2n+5$ for $n^2/4<e<n^2/3$ (upper_bound_p251) and credits it to "the construction described in [4]", his reference [4] being that same 1985 note. The value of $h(n)/n$ is not known; no source settling the "in particular" question was found in the search whose scope the Current assessment records, and the thread's June--July 2026 claims that the conjectured lower bound fails are a pending partial claim, unreviewed and recorded below. This is a bounded negative finding, not a certificate of openness. Erdős's 1982 report that the inequality $3n/2$ was proved by Edwards is recorded beside these bounds, with which it is inconsistent, and is not resolved here.

Source. erdosproblems.com/1033, accessed 2026-09-18T15:02Z: the problem page (labeled OPEN, the site's label for a problem that is open and not settled by a finite computation; last edited 3 April 2026; source keys [BoNi05], [Er82e], [Er93], [ErLa85], [Fa88], [Fa92]; commentary citing the same and Problem 904), its nine-comment discussion thread (27 June to 7 August 2026) and its empty proof-claim tab. Cite as: T. F. Bloom, Erdős Problem #1033, https://www.erdosproblems.com/1033, accessed 2026-09-18.

References.

• [Fa88] Fan, Genghua, Degree sum for a triangle in a graph. J. Graph Theory 12 (1988), no. 2, 249--263, doi:10.1002/jgt.3190120216; the definitions, pp. 249--250; the bounds and the § 2 construction, pp. 251--252; Theorem 1, p. 252; Corollary 1.1, p. 253; Theorem 2, p. 259; Theorem 3 and Corollary 3.1, pp. 261--262. Library home: fan_1988_degree_sum_triangle_graph; paged at theorem_1 and upper_bound_p251.
• [ErLa85] Erdős, P. and Laskar, R., A note on the size of a chordal subgraph. Congr. Numer. 48 (1985), 81--86 (the Rényi archive scan); the summary and the Edwards remark, p. 82; Theorem 1, p. 82; Theorem 2, p. 83. Library home: erdos_1985_note_size_chordal_subgraph; paged at theorem_1 and theorem_2.
• [ErLa83] Erdős, P. and Laskar, R., On maximum chordal subgraphs. Congr. Numer. 39 (1983), 367--373; [ErLa85]'s reference [8] and [BoNi05]'s reference [5]. Not a site key; not held.
• [BoNi05] Bollobás, Béla and Nikiforov, Vladimir, The sum of degrees in cliques. Electron. J. Combin. 12 (2005), N21, doi:10.37236/1988; cited from arXiv:math/0410218v1; the introduction, p. 2; Theorem 3, p. 8. Library home: bollobas_2005_sum_degrees_cliques; paged at theorem_3.
• [Fa92] Faudree, Ralph J., Complete subgraphs with large degree sums. J. Graph Theory 16 (1992), no. 4, 327--334, doi:10.1002/jgt.3190160406 (Crossref record with the publisher's abstract). Not held; the site's source for Erdős's construction, quoted here through [BoNi05].
• [Ed77] Edwards, C. S., The largest vertex degree sum for a triangle in a graph. Bull. London Math. Soc. 9 (1977), no. 2, 203--208, doi:10.1112/blms/9.2.203 (Crossref record accessed). Not a site key; [ErLa85]'s reference [7], whose theorem it states. Not held.
• [Er82e] Erdős, Paul, Some of my favourite problems which recently have been solved. Proceedings of the International Mathematical Conference (Singapore, 1981), North-Holland Math. Stud. 74 (1982), 59--79; §5, printed p. 71. Library home: erdos_1982_my_favourite_problems_which_recently_have.
• [Er75] Erdős, P., Some recent progress on extremal problems in graph theory. Congr. Numer. XIV (1975), 3--14; Chapter 4, printed p. 13, the conjecture for $e\ge n^2/3$. Not a site key for this problem. Library home: erdos_1975_recent_progress_extremal_problems_graph_theory; paged at conjecture_p13.
• [Er93] Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. 16 (1993), 333--350, doi:10.1080/16073606.1993.9631741. Chapter V, problem 3, printed pp. 343--344: the definition of $h(n)$, display (1) $2(\sqrt3-1)n\ge h(n)>\frac{21n}{16}$, the upper bound credited to Erdős and Laskar (its reference [45], which is [ErLa85]) and the lower to Fan (its [46], [Fa88]), and "Perhaps the upper bound in (1) is the correct value for $h(n)$"; no construction and no proof. Library home: erdos_1993_my_favorite_solved_unsolved_problems_graph_theory.

Formalization. None. No file ErdosProblems/1033.lean existed in formal-conjectures at main on 2026-09-18 (the directory FormalConjectures/ErdosProblems/, 673 entries, and the recursive tree, 1,742 entries, were listed in full), and none exists; the page of 2026-09-18 shows "Formalised statement? No (create one)"; the community database (teorth/erdosproblems, data/problems.yaml, fetched and again 2026-10-06) records the problem as open (last update 12 December 2025), unformalized, with no formalized statement and an OEIS entry marked "possible".

Current assessment

The question (site formulation of 2026-09-18T15:02Z). The statement above; OPEN; last edited 3 April 2026. The site's commentary, in this page's words: the problem is a conjecture of Bollobás and Erdős; [Er82e] asks whether $h(n)\ge\frac32n$; the bounds now known are $\frac{21}{16}n\le h(n)\le2(\sqrt3-1)n+O(1)$, the upper bound credited to Erdős and Laskar [ErLa85], the lower bound to Fan [Fa88], with the decimal values $1.464$ and $1.3125$ noted; the upper bound is said not to be explicit in [ErLa85], a paper about chordal subgraphs (subgraphs with no induced cycle on more than three vertices), whose bearing on the problem is that a triangle with all its incident edges is a chordal subgraph of $G$, and the construction is said to be made more explicit in [Fa88]; the construction follows, recomputed below; and the commentary ends with the general function $\Delta_r(n,m)$, Erdős's bound $\Delta_r(n,m)\le(1-\epsilon)2rm/n$ for $t_{r-1}(n)<m<t_r(n)-\delta n^2$, referred to [Fa92], and Bollobás and Nikiforov's $\Delta_r(n,m)\ge(1-\epsilon)2rm/n$ for $m>t_r(n)-\delta n^2$. The thread's nine comments are recorded below, the claimed refutation as a pending partial claim with its own page and the rest as leads; the proof-claim tab is empty; the community database record says open.

The lower bounds.

• [Fa88], Theorem 1 (p. 252; theorem_1): "Let $G\in\mathcal G(n;e)$. If $e>n^2/4$, then $\tau(G)>\frac{21e}{4n}$", where $\mathcal G(n;e)$ is the class of graphs with $n$ vertices and $e$ edges and $\tau(G)$ the largest degree sum of a triangle in $G$ (p. 250); Corollary 1.1 (p. 253) restates it as $f(n,e)>21e/4n$ for $f(n,e)=\min\{\tau(G):G\in\mathcal G(n;e)\}$, and the introduction (p. 251) draws "In particular, $f(n,\lfloor n^2/4\rfloor+1)>\frac{21}{16}n$". Since $f(n,\lfloor n^2/4\rfloor+1)$ is $h(n)$, this is $h(n)>21n/16$, strict as printed (the abstract prints a weak inequality). The paper frames the question as Bollobás and Erdős's, its [1], and says the bound improves Erdős and Laskar's $(1+\epsilon)n$, its [4], which its reference list (p. 263) identifies as [ErLa85]. The proof (p. 258) is an induction on $n$ from Lemma 2 (p. 255), which gives $\tau(G)\ge5e/n+\delta/4$ for minimum degree $\delta$ when $e>n^2/4$, through a covering of the vertices by complete graphs, double-triangles, a matching and a stable set (pp. 253--257). Theorem 2 (p. 259) gives a second lower bound, $\tau(G)\ge2n+4(\sqrt{e(4e-n^2)}-e)/n$, which the paper's Remark says exceeds $21e/4n$ for $e\ge0.26n^2$ and which is below $n+4$ at $e=\lfloor n^2/4\rfloor+1$, so it does not bear on $h(n)$. Read depth: claims checked; the proof of Theorem 1 from Lemma 2 followed; the proofs of Lemmas 1--2 and Theorem 2 read for structure.
• [ErLa85], Theorem 2 (p. 83; theorem_2): "Any graph $G(n,[\frac{n^2}4]+1)$ contains a chordal subgraph of at least $n(1+\varepsilon)$ edges if $n>n_0(\varepsilon)$ where $\varepsilon>0$ is a fixed positive number", proved, as the summary on p. 82 says, by showing "the existence of a tringle [sic] $xyz$, with $\deg x+\deg y+\deg z>n(1+\eta)$ for small $\eta>0$" (the triangle with its incident edges being the chordal subgraph). So $h(n)\ge(1+\eta)n$ for large $n$ and an unspecified $\eta>0$, the bound Fan's abstract says he improves (p. 249). Read depth: claims checked; the proof (pp. 83--85) read for structure.
• The regime $e\ge n^2/3$ is different: there Edwards's 1977 theorem, as [ErLa85] p. 82 states it ("any graph $G(n,m)$ with $m\ge n^2/3$ contains a triangle $xyz$, where $\deg x+\deg y+\deg z\ge2n$") and as [Fa88]'s Corollary 3.1 (p. 262, headed "(Edwards [3])") restates and reproves it ($\tau(G)\ge6e/n$ for $e\ge n^2/3$, with equality if and only if $G$ is regular), gives degree sum $2n$, the $r=3$ case of Problem 904 stated in [Er75], p. 13 (conjecture_p13).

The upper bound, recomputed. The site's construction, checked here as an authored computation. Let $m=\lfloor n^2/4\rfloor+1$, $k=cn$ and $l=n-k$ with $\frac13\le c\le1$. Take the complete bipartite graph $K_{k,l}$ and add $m-kl$ edges inside the $k$-side as a bipartite graph between two halves of that side; this is possible with every added degree at most $\lceil2(m-kl)/k\rceil$ when $m-kl\le k^2/4$, which holds since $m-kl=(\tfrac12-c)^2n^2+O(1)\le c^2n^2/4$ for $c\ge\frac13$. The graph has $n$ vertices and $m$ edges. The $l$-side is independent and the added graph is triangle-free, so every triangle has one vertex on the $l$-side (degree $k$) and two on the $k$-side (degree at most $l+\lceil2(m-kl)/k\rceil$ each), and its degree sum is at most $k+2l+2\lceil2(m-kl)/k\rceil=(c^{-1}+3c-2)n+O(1)$, using $2(m-kl)/k=(\tfrac1{2c}-2+2c)n+O(1/n)$. The function $c^{-1}+3c$ is minimized at $c=1/\sqrt3$, where it equals $2\sqrt3$, giving $h(n)\le(2\sqrt3-2)n+O(1)=2(\sqrt3-1)n+O(1)$; the site's formula and constant are reproduced. [Fa88], § 2 (pp. 251--252; upper_bound_p251), prints the same graph with $m=\lceil2\sqrt{3e}/3\rceil$ vertices on the side that receives the extra edges, and proves $f(n,e)<4\sqrt{3e}-2n+5$ for $n^2/4<e<n^2/3$, which at $e=\lfloor n^2/4\rfloor+1$ reads $2(\sqrt3-1)n+5+O(1/n)$ (a one-line substitution made here). Attribution: the site credits [ErLa85] and says the bound is not explicit there; the six pages of [ErLa85] carry no construction bounding a triangle's degree sum from above (their only extremal graph is the complete bipartite graph of Theorem 1, which has no triangle); Fan writes "we use the construction described in [4]" (p. 251), and his reference [4] (p. 263) is [ErLa85], not [ErLa83]. [Er93], Chapter V, problem 3, printed p. 344, states the bounds as display (1), "$2(\sqrt3-1)n\ge h(n)>\frac{21n}{16}$", and says "The upper bound is due to Renu Laskar and myself [45] and the lower bound is due to G. Fan [46] who improved significantly our lower bound $n(1+\epsilon)$. Perhaps the upper bound in (1) is the correct value for $h(n)$", its [45] being [ErLa85] (reference list, p. 349); so the survey makes the same attribution as the site and gives no construction either. The construction is therefore printed in [Fa88] § 2; the attribution to [ErLa85] by Fan, the survey and the site is not borne out by the note's six pages, and is recorded as printed.

The 1982 report, side by side with the bounds. [Er82e], §5, printed p. 71: Erdős recalls a conjecture made with Bollobás, that every $G(n;[\frac{n^2}4]+1)$ has an edge lying in at least $n/6$ triangles, best possible if true, and says that its proof needed a second conjecture, displayed as (1): for $m>\frac{n^2}4$, every $G(n;m)$ contains a triangle $(x_1,x_2,x_3)$ with $v(x_1)+v(x_2)+v(x_3)\ge\frac{3n}2$, $v(x)$ being the degree of $x$; a more general form, for $k(r)$ in place of $k(3)$, was also formulated. He then writes: "Edwards proved (1) and he in fact proved our conjecture nearly in its full generality", citing C. S. Edwards, Complete subgraphs with largest sum of vertex degrees, Coll. Math. Soc. J. Bolyai 18 (Combinatorics), North-Holland 1978, p. 293. Display (1) is $h(n)\ge3n/2$ for every $n$; the site reports [Er82e] as asking whether $h(n)\ge\frac32n$, while the page reports (1) as proved by Edwards. The upper bound above gives, for every large $n$, a graph with more than $n^2/4$ edges all of whose triangles have degree sum at most $2(\sqrt3-1)n+O(1)<3n/2$; so (1) as printed fails for large $n$, and the 1982 report that Edwards proved it cannot stand as printed. Which statement Erdős meant by (1) is not decided here: the Edwards theorem on triangle degree sums that the sources cited state is the one for $e\ge n^2/3$ with bound $2n$ ([ErLa85] p. 82; [Fa88] Corollary 3.1, p. 262), and Edwards's Bolyai paper, which the 1982 reference names, is not held. The three items, the 1982 wording, the contradiction and the current bounds, are recorded as printed and cited; the field open is not affected.

The general function. For $\Delta_r(n,m)$, the least maximal degree sum of an $r$-clique over graphs with $n$ vertices and $m$ edges, Problem 904 covers $m\ge t_r(n)$ (proved: $\Delta_r(n,m)\ge2rm/n$). In the range $t_{r-1}(n)<m<t_r(n)$, [BoNi05]'s introduction (p. 2) says the value "is essentially unknown even for $r=3$" and attests, through Faudree, Erdős's construction with $\Delta_r(n,m)\le(1-\varepsilon)2rm/n$ for $t_{r-1}(n)<m<t_r(n)-\delta n^2$; its Theorem 3 (p. 8) gives $\Delta_r(n,m)>(1-\varepsilon)2rm/n$ for $m>t_r(n)-\delta n^2$ and $n>n_0(\varepsilon)$, the stability the site quotes. At $r=3$ the theorem concerns edge counts within $\delta n^2$ of $n^2/3$ and says nothing about $m=\lfloor n^2/4\rfloor+1$, where $2rm/n=3n/2+O(1/n)$ and the bounds above place $h(n)$ between $1.3125n$ and $1.4641n+O(1)$. Read depth for [BoNi05]: claims checked; Theorem 3's proof read for structure.

The thread (a pending partial claim and leads, not status). Nine comments, from the discussion page as of 2026-09-18, none adopted into the site's statement or commentary (which are unchanged since 3 April 2026); the claimed refutation of the conjectured lower bound has a claim page under the write-up that publishes it, 2026_07_27_wouter_cvb:

• 27 June 2026 (the account rickyc): a claim that an explicit construction produced by GPT-5.5 Pro, as the post names the system, disproves the proposed lower bound, with a link to a chat transcript (not cited here).
• 26 July 2026 (the account Johan Land): a report that the construction holds up and that the conjectured lower bound $h(n)\ge(2(\sqrt3-1)-o(1))n$ is false, giving $\frac{21}{16}n\le h(n)\le1.46393n+O(1)$, and noting that Fan's lower bound has not moved since 1988 and that no candidate for the true constant remains; a reply the same day (rickyc) expects stronger constructions.
• 27 July 2026 (the account Wouter CvB): the part densities of the construction, a blow-up of the butterfly graph, optimized with further AI assistance (no system named) to $1.463877226\ldots$, a root of a polynomial of degree $7$, with a linked three-page write-up under the author line Wouter CvB, dated 27 July 2026 (accessed; its Lemma 1 states the bound $1.463878\,n$, its Lemma 2 the root $y_0=1.463877226961\ldots$ of a degree-$7$ polynomial, and it says blow-ups of all graphs on at most $8$ vertices were tried) and the post's remark that blow-ups of all connected graphs on at most seven vertices were tried and none beats the butterfly; two replies the same day (rickyc) report that GPT confirms the computation and judges the construction hard to beat, so that it may be optimal.
• 1 and 7 August 2026 (the account RealBelgian): Fan's lower bound improved by a flag-algebra computation to about $1.45016$ and then to about $1.4525$, described as probably still not sharp, with a note or paper promised; between them (1 August, rickyc) a report that an extended search with GPT (an eight-hour run of what the post calls Sol Ultra, with four subagents) could not beat the 27 July construction and proved it optimal within several restricted classes of constructions.

If the construction holds, the "in particular" question has answer no and $h(n)/n$ would lie between about $1.4525$ and $1.463877$; none of this is checked here, no paper or preprint carrying it was found, and the site has not adopted it. The write-up is the publication carrying the refutation claim and has the claim page (claimed, partial); the 27 June 2026 thread claim (rickyc), a thread post with a chat transcript and no manuscript, is disclosed on that page and has no page of its own; the flag-algebra improvements of the lower bound answer neither question of the statement and stay leads here. The AI systems are named as the posts name them; no chat transcript is cited.

Search scope. None of the routes below found a refereed source improving either bound or settling the "in particular" question; the discussion thread carries the claimed refutation of the conjectured lower bound and its write-up of 27 July 2026, recorded above and on the claim page.

• The site: problem page, discussion thread and proof-claim tab as of 2026-09-18; the formal-conjectures directory and tree at main that day (no file 1033); the community database entry as fetched that day.
• Crossref: the records of doi:10.1002/jgt.3190120216 ([Fa88], with its abstract) and doi:10.1002/jgt.3190160406 ([Fa92], with its abstract); a bibliographic query for Edwards's 1977 title (Bull. London Math. Soc. 9 (1977), 203--208) and one for [BoNi05] (the EJC note, DOI 10.37236/1988).
• One paced request to the publisher's page for [Fa88] (HTTP 403, a script challenge page).
• Semantic Scholar: the citing papers of [Fa88] (seven records: [BoNi05], [Er93], a 1992 Discrete Mathematics paper on odd cycles, two 1988--1989 papers on unavoidable subgraphs with large degrees, and two unrelated; titles and venues only) and of [BoNi05] (one record, "Maximal chordal subgraphs", Combin. Probab. Comput. 2023, arXiv:2205.08474, on the Erdős--Laskar chordal-subgraph function; title only).
• arXiv API: the search all:"degree sum" AND all:triangle sorted by date (seven records, on Hamiltonicity and double stars, none on $h(n)$) and all:"sum of degrees" AND all:clique (one record, [BoNi05]); the record of math/0410218 (v1 only).
• The primary sources, at the pages cited: [ErLa85] pp. 81--86; [BoNi05] pp. 1--3 and 6--8; [Er82e] p. 71 and [Er75] p. 13.

Not searched: MathSciNet, zbMATH, Google Scholar, X; the flag-algebra computations were not read, and the thread's linked write-up was accessed after the search (its claim page). Not held: [Fa92], [Ed77], [ErLa83], Edwards's Bolyai 18 paper. The search did not cover [Er93] or [Fa88], which are cited above at pp. 343--344 and in full.

Remaining gaps. (1) [Fa88]'s lower bound is paged with its exact statement and proof pointer at theorem_1, the proof of Theorem 1 followed from Lemma 2 and the lemmas read for structure. (2) The construction behind the upper bound is printed in [Fa88] § 2 and paged at upper_bound_p251; what remains is the attribution: Fan, [Er93] and the site credit it to [ErLa85], whose six pages contain no such construction, and no earlier text printing it is held. (3) The 1982 report of (1) is inconsistent with the upper bound and is not resolved. (4) The thread's claimed refutation of the conjectured lower bound is a pending partial claim with its own page, 2026_07_27_wouter_cvb, resting on an unreviewed three-page write-up, and the flag-algebra improvements are unreviewed leads; neither has a paper. (5) [Fa92] is not held; [Er93], pp. 343--344, states both bounds with the site's attributions and the guess that the upper bound is the true value, without a construction. (6) Proof coverage: [ErLa85]'s two theorems and [BoNi05]'s Theorem 3 at claims checked; [Fa88]'s Theorem 1 at proof followed from Lemma 2, its lemmas and Theorem 2 at structure; the site's construction is recomputed here and followed in [Fa88] § 2. (7) There is no Lean statement of the problem.

Known results

• Fan 1988, Theorem 1 (refereed): $h(n)>21n/16$, from $\tau(G)>21e/4n$ for every graph with $n$ vertices and $e>n^2/4$ edges; the best lower bound in the sources read.
• Erdős--Laskar 1985, Theorem 2: $h(n)\ge(1+\eta)n$ for a fixed $\eta>0$ and large $n$; with Theorem 1 on chordal subgraphs of size $n$.
• Fan 1988, § 2 (the site's construction, recomputed above; credited to Erdős and Laskar by Fan and by the site): $f(n,e)<4\sqrt{3e}-2n+5$ for $n^2/4<e<n^2/3$, so $h(n)\le2(\sqrt3-1)n+O(1)$.
• [Ed77] (1977, not held), per [ErLa85] and reproved as [Fa88]'s Corollary 3.1 (p. 262): degree sum $\ge6e/n\ge2n$ once $e\ge n^2/3$, with equality only for regular graphs; a different regime.
• Bollobás--Nikiforov, Theorem 3 (2005, refereed): the stability bound near $t_r(n)$; Erdős's construction for $t_{r-1}(n)<m<t_r(n)-\delta n^2$, second-hand.
• [Er82e], p. 71: the conjecture (1) with $3n/2$, reported proved by Edwards and inconsistent with the upper bound.
• [Er93], pp. 343--344: the bounds $2(\sqrt3-1)n\ge h(n)>21n/16$ as stated in 1993, credited to Erdős--Laskar and Fan, with the guess that the upper bound is the true value.

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_1982_my_favourite_problems_which_recently_have
• bollobas_2005_sum_degrees_cliques
• bollobas_2005_sum_degrees_cliques / theorem_3
• erdos_1985_note_size_chordal_subgraph
• erdos_1985_note_size_chordal_subgraph / theorem_1
• erdos_1985_note_size_chordal_subgraph / theorem_2
• erdos_1993_my_favorite_solved_unsolved_problems_graph_theory
• fan_1988_degree_sum_triangle_graph
• fan_1988_degree_sum_triangle_graph / lemma_2
• fan_1988_degree_sum_triangle_graph / theorem_1
• fan_1988_degree_sum_triangle_graph / theorem_2
• fan_1988_degree_sum_triangle_graph / theorem_3
• fan_1988_degree_sum_triangle_graph / upper_bound_p251