Erdős problem 80 — wave 5h report
Date: 2026-07-26 UTC
Claim labels used throughout:
- [a] elementary-rigorous — proved below from definitions or by a finite
certificate whose verification is transparent.
- [b] rigorous-modulo-named-theorem — the deduction is rigorous assuming
the explicitly cited theorem.
- [c] plausible/structural-unverified — a proposed route or an honest
literature-search conclusion, not a theorem.
- [d] computational-only — an exhaustive finite computation, not promoted
to an asymptotic theorem.
0. Mandatory live-page check
I accessed the live page through the Bright Data browser route on
2026-07-26. I separately extracted the rendered problem page, its LaTeX
view, its links and status fields, and the linked discussion thread. A
full-page screenshot agreed with the extracted text. The page says it was
last edited on 07 April 2026.
Verbatim live statement
> Let \(c>0\) and let \(f_c(n)\) be the maximal \(m\) such that every graph
> \(G\) with \(n\) vertices and at least \(cn^2\) edges, where each edge is
> contained in at least one triangle, must contain a book of size \(m\),
> that is, an edge shared by at least \(m\) different triangles.
>
> Estimate \(f_c(n)\). In particular, is it true that
> \(f_c(n)>n^{\epsilon}\) for some \(\epsilon>0\)? Or
> \(f_c(n)\gg \log n\)?
Live sources: problem page,
Verbatim listed results
> A problem of Erdős and Rothschild. Alon and Trotter showed that, provided
> \(c<1/4\), \(f_c(n)\ll_c n^{1/2}\). Szemerédi observed that his regularity
> lemma implies that \(f_c(n)\to \infty\).
>
> Edwards (unpublished) and Khadzhiivanov and Nikiforov [KhNi79] proved
> independently that \(f_c(n) \geq n/6\) when \(c>1/4\) (see [905]).
>
> Fox and Loh [FoLo12] proved that
> \[ > f_c(n) \leq n^{O(1/\log\log n)} > \]
> for all \(c<1/4\), disproving the first conjecture of Erdős.
>
> The best known lower bounds for \(f_c(n)\) are those from Szemerédi's
> regularity lemma, and as such remain very poor.
>
> See also [600] and the entry in the graphs problem collection.
The displayed references identify:
[FoLo12]: Jacob Fox and Po-Shen Loh, *On a problem of Erdős and
Rothschild on edges in triangles*, Combinatorica (2012), 619–628.
[KhNi79]: N. G. Khadzhiivanov and S. V. Nikiforov, *Solution of a
problem of P. Erdős about the maximum number of triangles with a common
edge*, C. R. Acad. Bulgare Sci. (1979), 1315–1318.
Sole live comment, verbatim
The page reports one comment. Its full mathematical text is:
> [Po18] by A. Potechin is another partial improvement on this problem.
>
> Moreover, this problem is also reported to be in [Er88], [Er92] and
> [Er98]
>
> [Er88]: P. Erd˝os, Problems and results in combinatorial analysis and
> graph theory, Proceedings of the First Japan Conference on Graph Theory
> and Applications (Hakone, 1986), Discrete Math. 72 (1988), p. 81-92.
>
> [Er92]: P. Erd˝os, Some of my favourite problems in various branches of
> combinatorics. Combinatorics 92 (Catania, 1992). Matematiche (Catania)
> 47 (1992), no. 2, p. 231-240
It was posted by Alfaiz at 12:26 on 04 May 2026. The rendered [Po18]
link points to https://arxiv.org/pdf/1412.1838.
Status and collaboration/proof markers
- [b] Status badge: OPEN.
- [b]
1 comment on this problem. - [b]
0 claimed proofs for this problem. - [b]
Likes this problem: None. - [b]
Interested in collaborating: None. - [b]
Currently working on this problem: None. - [b]
This problem looks difficult: None. - [b]
This problem looks tractable: None. - [b]
The results on this problem could be formalisable: None. - [b]
I am working on formalising the results on this problem: None. - [b]
Formalised statement? No.
Thus the mandatory stop condition did not fire: there is no claimed proof
and no current worker.
1. Primary-source literature audit
Original problem
[b] The original six-page Erdős paper exists in the Rényi archive:
[P. Erdős, Some problems on finite and infinite graphs (1987),
PDF](https://www.renyi.hu/~p_erdos/1987-28.pdf). On printed page 226,
Problem 11 names Bruce Rothschild, defines \(f(n;c)\), records the Alon
and Szemerédi observations, and asks whether \(f(n;c)>n^\epsilon\), or at
least \(f(n;c)>\log n\). The downloaded primary PDF has SHA-256
b22809ce28667517eb39249bf54bff2140c548475306070989f3f8964870cb71.
The optional online mode of the checker downloads this PDF afresh, checks
that digest, and checks the OCR for the stable identifying tokens.
Fox–Loh upper construction
[b] The cited paper and arXiv identifier both exist:
- Jacob Fox and Po-Shen Loh, *On a problem of Erdős and Rothschild on
edges in triangles*, Combinatorica 32 (2012), 619–628,
DOI 10.1007/s00493-012-2844-3,
Its Theorem 1.1 gives, for all sufficiently large \(n\), an \(n\)-vertex
graph with
\[ \frac{n^2}{4}\left(1-e^{-(\log n)^{1/6}}\right) \]edges, every edge in a triangle, and no edge in more than
\(n^{14/\log\log n}\) triangles. Consequently, for every fixed
\(c<1/4\),
\[ f_c(n)\le n^{O(1/\log\log n)}. \]The paper also explicitly says that Fox's improved graph-removal bound
gives
\[ f_c(n)\ge 2^{\Omega_c(\log^* n)} \]for every fixed \(c>0\). This is still vastly below \(\log n\).
Potechin's omitted threshold-regime improvement
[b] The live comment's [Po18] link is a real primary source:
Aaron Potechin, A note on a problem of Erdos and Rothschild,
arXiv:1412.1838. The arXiv record says
it was submitted on 4 December 2014; [Po18] is the website comment's
citation label, not evidence of a 2018 journal publication. I found no
journal record and do not invent one.
Potechin defines the threshold-regime quantity with
\[ e(G)=\frac{n^2}{4}-nf(n). \]His Theorem 1.3 states that, when \(f(n)\le n/1000\), either
\(bk(G)>n/1000\), or
\[ f(n)\bigl(f(n)+bk(G)\bigr)bk(G)\ge \frac{n^2}{1250}. \]Corollary 1.4 gives the explicit lower bound
\[ bk(G)\ge \min\left\{ \frac{n}{50\sqrt{f(n)}}, \frac{n^2}{2500f(n)^2}, \frac{n}{1000} \right\} \]under the paper's stated integrality hypothesis. This is a genuine
improvement near the \(n^2/4\) threshold. It does not settle fixed
\(c<1/4\), where \(f(n)\) is linear in \(n\).
Later explicit reformulation
[b] Noga Alon and Adi Shraibman,
*Algorithmic Number On the Forehead Protocols Yielding Dense
Ruzsa–Szemerédi Graphs and Hypergraphs*,
arXiv:2001.00387, gives explicit
communication-protocol versions of the dense Ruzsa–Szemerédi and
Fox–Loh-type constructions. In the graph case it recovers the same
\(n^{O(1/\log\log n)}\) scale; it does not improve the open logarithmic
lower bound.
Search conclusion
[c] I searched the exact title, the notations \(h(n,c)\),
\(\gamma(n,f)\), and \(f_c(n)\), the phrases “every edge is in/contained
in a triangle” with “booksize”, the Fox–Loh citation trail, and arXiv
records through July 2026. I found Potechin and Alon–Shraibman, but no
primary source claiming either \(f_c(n)\gg\log n\) or a counterexample to
it. This is an honest search result, not a proof that no uncatalogued
paper exists. It agrees with the live page's 2026 OPEN status.
The standalone checker has an --online-citations mode which independently
validates the arXiv metadata and abstract claims above, the Fox–Loh
Crossref record, and the original Erdős PDF.
2. A finite extremal formulation
For a graph \(G\), write
\[ b(G)=\max_{uv\in E(G)}|N(u)\cap N(v)| \]for its booksize, with \(b(\varnothing)=0\). Define
\[ M_k(n)=\max\left\{ e(G): |V(G)|=n,\ \text{every edge of }G\text{ is in a triangle},\ b(G)\le k \right\}. \tag{2.1} \]Also let
\[ F(n,E)=\min\left\{ b(G): |V(G)|=n,\ e(G)\ge E,\ \text{every edge is in a triangle} \right\}. \tag{2.2} \][a] Whenever \(1\le E\le\binom n2\),
\[ F(n,E)=\min\{k:E\le M_k(n)\}. \tag{2.3} \]Indeed, \(E\le M_k(n)\) supplies a witness with at least \(E\) edges and
booksize at most \(k\). If \(E>M_{k-1}(n)\), no qualifying graph with at
least \(E\) edges can have booksize at most \(k-1\).
Therefore, with
\[ E_c(n)=\lceil cn^2\rceil, \][a]
\[ f_c(n)=F(n,E_c(n)) \tag{2.4} \]provided \(E_c(n)\le\binom n2\). If \(E_c(n)>\binom n2\), there is no
qualifying simple graph, so the universal condition is vacuous and there
is no maximal finite \(m\).
3. A reusable degree-sum lemma
Lemma
[a] If \(G\) has \(n\) vertices, \(e>0\) edges, and
\(b(G)\le b\), then
\[ \sum_{v\in V(G)}d(v)^2\le e(n+b) \tag{3.1} \]and consequently
\[ e\le \frac{n(n+b)}4. \tag{3.2} \]Proof
For an edge \(uv\), let
\[ q=|N(u)\cap N(v)| \]and let \(r\) be the number of vertices adjacent to neither \(u\) nor
\(v\). Partitioning the other \(n-2\) vertices gives
\[ d(u)+d(v)-n=q-r\le q\le b. \]Summing over all edges,
\[ \sum_v d(v)^2 =\sum_{uv\in E(G)}(d(u)+d(v)) \le e(n+b). \]Cauchy–Schwarz and \(\sum_vd(v)=2e\) give
\[ \frac{4e^2}{n}\le\sum_vd(v)^2\le e(n+b), \]which yields (3.2). \(\square\)
4. Exact solution of the complete \(n=8\) finite problem
Theorem
[a] For \(n=8\),
\[ \boxed{ \bigl(M_0(8),M_1(8),\ldots,M_6(8)\bigr) =(0,12,18,21,24,24,28). } \tag{4.1} \]The proof below is elementary. The checker independently exhausts the
two delicate upper bounds \(k=1,2\).
Sharp constructions
Each row is directly checked from its description.
| book cap \(k\) | graph | edges | actual booksize |
|---:|---|---:|---:|
| 1 | four triangles in a loose \(4\)-cycle | 12 | 1 |
| 2 | \(\overline K_2\vee C_6\) | 18 | 2 |
| 3 | \(K_{2,3,3}\) | 21 | 3 |
| 4 | \(K_8\) minus a perfect matching | 24 | 4 |
| 5 | \(K_8\) minus a perfect matching | 24 | 4 |
| 6 | \(K_8\) | 28 | 6 |
For the first row, one explicit set of triangles is
\[ 012,\quad 234,\quad 456,\quad 670. \]For \(\overline K_2\vee C_6\), a cycle edge has the two apices as its
common neighbors, and an apex–cycle edge has the two cycle neighbors of
its cycle endpoint. Thus every edge has codegree exactly \(2\).
In a complete tripartite graph, an edge between two parts has the third
part as its common neighborhood, so \(K_{2,3,3}\) has booksize \(3\).
In \(K_8\) minus a perfect matching, an edge's endpoints have two
distinct forbidden partners, leaving exactly four common neighbors.
It remains to prove the upper bounds.
Cap \(k=1\): \(M_1(8)\le12\)
Suppose every edge lies in exactly one triangle. Then graph triangles
are edge-disjoint, and
\[ e(G)=3|\mathcal T|. \]If \(e(G)\ge13\), then \(e(G)\ge15\), so choose five graph triangles.
Regard them as the blocks of a linear \(3\)-uniform hypergraph
\(\mathcal H\) on at most eight points. It has no Berge triangle: if
three blocks met pairwise in three distinct points, those intersection
points would form an additional graph triangle, putting its edges in
two triangles.
I now prove that no such \(\mathcal H\) exists. Let its used-point
degrees be \(r_1,\ldots,r_v>0\). There are \(15\) incidences. Linearity
gives
\[ \sum_i\binom{r_i}{2}\le\binom52=10. \tag{4.2} \]If \(v\le6\), convexity makes the left side at least \(12\), so
\(v\in\{7,8\}\). The positive integer partitions of \(15\) satisfying
(4.2) are exactly:
| \(v\) | point-degree multiset | \(\sum\binom{r_i}{2}\) |
|---:|---|---:|
| 7 | \(3^2\,2^4\,1\) | 10 |
| 7 | \(3\,2^6\) | 9 |
| 8 | \(3^3\,2\,1^4\) | 10 |
| 8 | \(4\,2^4\,1^3\) | 10 |
| 8 | \(3^2\,2^3\,1^3\) | 9 |
| 8 | \(3\,2^5\,1^2\) | 8 |
| 8 | \(2^7\,1\) | 7 |
Build the intersection graph \(J\) on the five blocks. A point of
degree \(r\) contributes a clique \(K_r\), and these cliques are
edge-disjoint. Every triangle of \(J\) must lie in one of these
designated cliques; otherwise it is a Berge triangle.
The seven rows are impossible:
1. For \(2^7\,1\), \(J\) would be a triangle-free five-vertex graph with
seven edges, contradicting Mantel's elementary six-edge bound.
2. For \(3\,2^5\,1^2\), \(J\) has eight edges but only one permitted
triangle. Any eight-edge graph on five vertices is \(K_5\) minus two
edges and has at least \(10-2\cdot3=4\) triangles.
3. For \(3\,2^6\), \(J=K_5-e\) has seven triangles but only one
designated \(K_3\).
4. In either row with two \(K_3\)'s, the edge-disjoint triangles on five
block-vertices meet in one block. Every additional edge between the
two opposite pairs completes a non-designated triangle through that
common block. Yet the row requires three or four such edges.
5. For \(3^3\,2\,1^4\), three triangles and one edge would partition
\(E(K_5)\). Triangle contributions give even degree at every vertex,
while the final edge changes parity at its two endpoints, impossible
since all \(K_5\) degrees are even.
6. For \(4\,2^4\,1^3\), after the designated \(K_4\), the four remaining
edges join the fifth block to each \(K_4\) block. They correspond to
four distinct intersection points on that fifth block, impossible
for a three-element block.
This contradiction proves \(e(G)\le12\). The loose \(4\)-cycle shows
equality.
The checker also verifies this bound independently by inspecting all
\[ \binom{56}{5}=3{,}819{,}816 \]five-triangle families on eight labelled vertices. Exactly 84,000
families are edge-disjoint, and every one has an additional triangle
in its edge union.
Cap \(k=2\): \(M_2(8)\le18\)
The degree-sum lemma first gives \(e(G)\le20\). Suppose
\(e(G)\in\{19,20\}\), and let \(H=\overline G\). For a nonedge \(uv\)
of \(H\), equivalently an edge of \(G\),
\[ |N_G(u)\cap N_G(v)| =6-|N_H(u)\cup N_H(v)|. \]Every \(G\)-edge is in one or two triangles, so
\[ |N_H(u)\cup N_H(v)|\in\{4,5\} \quad\text{for every nonedge }uv\text{ of }H. \tag{4.3} \]First, \(\delta(H)\ge2\). A degree-zero vertex and each of its seven
nonneighbors force all seven other degrees to be at least \(4\), far
above either available degree sum. If \(d_H(u)=1\), then each of the six
vertices other than \(u\) and its neighbor has degree at least \(3\) by
(4.3), again forcing degree sum at least \(20\).
If \(e(G)=20\), then \(e(H)=8\), so every \(H\)-degree is \(2\). The two
neighbors of any vertex share that vertex. If they were nonadjacent,
their neighborhood union would have size at most \(3\), contradicting
(4.3). Hence every component of the \(2\)-regular graph \(H\) is a
triangle, impossible on eight vertices.
If \(e(G)=19\), then \(e(H)=9\). Since all degrees are at least \(2\)
and sum to \(18\), the degree multiset is one of
\[ (4,2,2,2,2,2,2,2), \qquad (3,3,2,2,2,2,2,2). \tag{4.4} \]For any vertex \(w\), its degree-two neighbors form a clique: two
nonadjacent such neighbors would share \(w\) and have neighborhood
union at most \(3\), violating (4.3). Such a clique has at most two
vertices, because each already uses one edge to \(w\).
The first sequence in (4.4) is therefore impossible: its degree-four
vertex has four degree-two neighbors. In the second sequence, call the
degree-three vertices \(x,y\). Each has at most two degree-two
neighbors, hence \(xy\in E(H)\), and each has exactly two low-degree
neighbors. Those two pairs must be disjoint; a shared low neighbor
would already have degree two from \(x,y\), but cliquehood at \(x\)
would force a third edge. Each pair is itself adjacent and therefore
saturated. The two remaining degree-two vertices can then only meet
each other, leaving them with degree one. This final contradiction
proves \(e(G)\le18\).
As an independent audit, the checker generates every labelled
complement with the degree sequences surviving the argument:
- 3,507 graphs with degree sequence \(2^8\);
- 21,000 graphs over all labelings of \(4,2^7\);
- 119,420 graphs over all labelings of \(3,3,2^6\).
None satisfies (4.3).
Cap \(k=3\): \(M_3(8)\le21\)
The degree-sum lemma gives \(e\le22\). If \(e=22\), the degree sum is
\(44\). Among eight integer degrees with this sum, the minimum square
sum is attained by four \(5\)'s and four \(6\)'s:
\[ \sum_vd(v)^2\ge4\cdot25+4\cdot36=244. \]But (3.1) gives
\[ \sum_vd(v)^2\le22(8+3)=242, \]a contradiction. Thus \(e\le21\), sharply attained by \(K_{2,3,3}\).
Cap \(k=4\): \(M_4(8)\le24\)
Equation (3.2) immediately gives
\[ e\le\frac{8(8+4)}4=24, \]sharply attained by \(K_8\) minus a perfect matching.
Cap \(k=5\): \(M_5(8)\le24\)
Equation (3.2) first gives \(e\le26\). If \(e=25\) or \(26\), then
\(\overline G\) has only three or two edges. Those missing edges touch
at most six vertices, so at least two vertices are isolated in
\(\overline G\). Their edge in \(G\) has all other six vertices as
common neighbors, giving a book of size \(6\), contrary to the cap.
Thus \(e\le24\), again attained by \(K_8\) minus a perfect matching.
Cap \(k=6\)
The complete graph \(K_8\) has 28 edges and booksize 6, and no simple
eight-vertex graph has more edges.
This completes the proof of (4.1).
Exact value of \(f_c(8)\)
Combining (2.3) with (4.1), [a]
\[ F(8,E)= \begin{cases} 1,&1\le E\le12,\\ 2,&13\le E\le18,\\ 3,&19\le E\le21,\\ 4,&22\le E\le24,\\ 6,&25\le E\le28. \end{cases} \tag{4.5} \]In particular, there is no edge threshold for which the forced value is
exactly \(5\).
Since \(E=\lceil64c\rceil\), [a]
\[ \boxed{ f_c(8)= \begin{cases} 1,&0the defining universal class is empty.
5. Independent exact census for \(3\le n\le7\)
The checker uses two independent exhaustive methods.
1. Union generation. Start with \(\{0\}\) and repeatedly adjoin the
bitwise union with each triangle edge-mask. This produces exactly all
unions of triangles.
2. All-mask scan. Inspect all \(2^{\binom n2}\) labelled graphs,
retain a graph iff the union of its present triangles equals its full
edge set, and compute all edge codegrees directly from bit-set
neighborhoods.
[a] The completeness observation is exact: every edge of \(G\) lies
in a triangle iff \(G\) is the union of its triangles. Thus the first
method omits no admissible graph, and the second is an independent
cross-check.
Both methods return the following frontiers. These rows are
[d] computational-only for \(n\le7\); the \(n=8\) row is the
elementary theorem proved above.
| \(n\) | \(M_0(n),M_1(n),\ldots,M_{n-2}(n)\) |
|---:|---|
| 3 | \(0,3\) |
| 4 | \(0,3,6\) |
| 5 | \(0,6,8,10\) |
| 6 | \(0,6,12,12,15\) |
| 7 | \(0,9,15,16,18,21\) |
| 8 | \(0,12,18,21,24,24,28\) |
The exact numbers of admissible labelled graphs by exact booksize are:
| \(n\) | booksize \(0,1,\ldots,n-2\) |
|---:|---|
| 3 | \(1,1\) |
| 4 | \(1,4,7\) |
| 5 | \(1,25,110,51\) |
| 6 | \(1,120,2487,2796,711\) |
| 7 | \(1,1155,78956,193396,118692,17725\) |
The total admissible labelled graph counts are
\[ 2,\ 12,\ 187,\ 6115,\ 409925 \]for \(n=3,4,5,6,7\), respectively. The two methods also agree on a
SHA-256 digest of the full increasing mask list at each \(n\), not only
on these aggregate statistics.
6. Exact reduction to the quantitative triangle-removal bottleneck
Let \(\delta_\triangle(\varepsilon)\) denote any valid triangle-removal
function in the following convention:
> Every \(n\)-vertex graph requiring deletion of at least
> \(\varepsilon n^2\) edges to become triangle-free contains at least
> \(\delta_\triangle(\varepsilon)n^3\) triangles.
Suppose \(G\) has \(e(G)\ge cn^2\), every edge lies in a triangle, and
\(b(G)\le b\). Let \(T\) be its number of triangles and let
\(\tau_\triangle(G)\) be the minimum number of edges meeting every
triangle.
[a] Since every edge is covered,
\[ 3T=\sum_{uv\in E(G)}|N(u)\cap N(v)|\ge e(G). \]Since a single edge meets at most \(b\) triangles, every triangle edge
cover has size at least
\[ \tau_\triangle(G)\ge\frac{T}{b} \ge\frac{e(G)}{3b} \ge\frac{c}{3b}n^2. \tag{6.1} \]On the other hand,
\[ 3T\le b\,e(G)\le b\binom n2<\frac{bn^2}{2}, \]so \(T Applying the triangle-removal lemma at \(\varepsilon=c/(3b)\) gives the exact implicit obstruction Thus any \(b\) for which the reverse strict inequality holds is too small to be the booksize of a qualifying graph. [b] Fox's quantitative graph-removal theorem, substituted into (6.2), yields the known \(2^{\Omega_c(\log^*n)}\) lower bound. Its tower dependence does not reach \(\log n\). [a] A bound of the form for all sufficiently small \(\varepsilon\) would be enough. Indeed, put \(b=\alpha\log n\). Then (6.3) gives Choosing, for example, \(\alpha \(n^{-1/2}\), which is eventually larger than \(b/(6n)=O(\log n/n)\), contradicting (6.2). Hence (6.3) would imply \(f_c(n)\gg_c\log n\). [c] I do not prove (6.3), and do not claim that a general removal bound of precisely this strength is necessary. The exact missing input for this standard route is the scale-specific inequality A theorem tailored to triangle-covered, bounded-book graphs could also supply it without improving the full triangle-removal lemma. This identifies the genuine uniformity gap: the finite frontiers and explicit \(n=8\) solution do not give a bound uniform in \(n\). In particular, the \(b=1\) case is already tied to the Ruzsa–Szemerédi \((6,3)\) phenomenon; finite exclusion is not the asymptotic density statement. The standalone checker is It uses only the Python standard library. Run: On this VM it completed all finite checks in about 20 seconds. It: independent scan of every labelled graph mask; bound; classes relevant to \(M_2(8)\); To re-fetch and validate the primary-source citation records as well, run: That mode completed in about 23 seconds and reported every citation audit PASS. I did not perform a naive \(n=9\) all-graph scan. It would inspect masks. Scaling the measured \(n=7\) scan gives roughly 109 single-core hours at the same per-mask throughput, about 4.5 core-days (roughly \$5.50–\$16.50 at \$0.05–\$0.15/core-hour). Since a nine-vertex mask has more triangles to inspect, this is optimistic. It would still be only a finite result. Structural generation or certified SAT would be the appropriate next computational method. The asymptotic logarithmic question remains open. The concrete progress is a complete, elementary exact solution at \(n=8\), explicit extremal graphs at every book cap, an independently reproduced \(n\le7\) census, and the precise quantitative triangle-removal inequality that the standard lower-bound route must beat. PARTIAL: Proved the exact frontier \(M_k(8)=(0,12,18,21,24,24,28)\), independently verified the full \(n\le7\) census, and isolated the removal-lemma scale needed for \(f_c(n)\gg\log n\); the asymptotic problem remains open.What would make the logarithmic bound follow by this route
7. Reproducibility and cost
runs/erdos80_wave5h_verify.py.python runs/erdos80_wave5h_verify.py
python runs/erdos80_wave5h_verify.py --online-citations
8. Outcome