-
Notifications
You must be signed in to change notification settings - Fork 1
/
problems.html
executable file
·763 lines (737 loc) · 28.9 KB
/
problems.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 TRANSITIONAL//EN">
<html>
<head>
<title>DR's Open Math Problems</title>
<!-- Updated 2020-05-20 -->
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<link rel="stylesheet" type="text/css" href="rorabaugh.css" />
<!-- Global site tag (gtag.js) - Google Analytics -->
<script
async
src="https://www.googletagmanager.com/gtag/js?id=G-8VSF4BT9C8"
></script>
<script>
window.dataLayer = window.dataLayer || [];
function gtag() {
dataLayer.push(arguments);
}
gtag("js", new Date());
gtag("config", "G-8VSF4BT9C8");
</script>
<script
src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"
type="text/javascript"
></script>
</head>
<body lang="EN-US">
<div id="menu">
<ul>
<li><a href="index.html">Home</a></li>
<li><a href="research.html">Research</a></li>
<li><strong>Open Problems</strong></li>
<li><a href="teaching.html">Teaching</a></li>
<li><a href="coding.html">Coding</a></li>
<li><a href="spanish.html">Español</a></li>
</ul>
</div>
<!--Open Problems-->
<div style="margin-top: 60px">
<h3>My Favorite Open Problems in Mathematics</h3>
<b>... on Graphs</b>
<ul>
<li>
A graph with m edges is <i>antimagic</i> provided there is an
injective labeling of the edges with \(\{1,2,...,m\}\) such that, if
each vertex is assigned the sum of the labels of the incident edges,
the vertex sums are pairwise distinct.
<br />
<b>Conjecture:</b> "Every connected graph different from \(K_2\) is
antimagic."
<br />
This was posed by Hartsfield and Ringle [<a
href="http://proofits.files.wordpress.com/2012/09/0123285526_graphtheory.pdf"
target="_blank"
rel="noopener noreferrer"
>1990</a
>, pp 108].
<br />
A natural generalization is: A graph with \(m\) edges is
<i>\(k\)-antimagic</i> provided there is an injective labeling of the
edges with \(\{1,2,...,m+k\}\) such that... the vertex sums are
pairwise distinct.
<br />
<b>Question:</b> For what \(k\) is every connected graph, except
\(K_2\), \(k\)-antimagic?
<br />
Berikkyzy et al. [<a
href="http://sites.google.com/site/rmgpgrwc/research-papers"
target="_blank"
rel="noopener noreferrer"
>2014+</a
>] proved that every connected graph on \(n \geq 3\) vertices is
\(\left\lfloor \frac{4n}{3} \right\rfloor\)-antimagic. <br /><br />
</li>
<li>
A <i>pseudograph</i> is an undirected graph with loops and multiple
edges allowed. An <i>\(\{r-t,t\}\)-factor</i> of an \(r\)-regular
graph is a spanning subgraph in which every vertex has degree either
\(r-t\) or \(t\). Assume henceforth that \(0 < t \leq \frac{r}{2}\).
<br />
Akbari and Kano [<a
href="http://dx.doi.org/10.1007/s00373-013-1324-x"
target="_blank"
rel="noopener noreferrer"
>2014</a
>] showed that if \(r > 4\) is odd and either \(t\) is even or \(t
\geq \frac{r}{3}\) is odd, then every \(r\)-regular pseudograph has an
\(\{r-t,t\}\)-factor. They further conjectured this to be the case for
odd \(r\) and all \(t\).
<br />
Axenovich and Rollin [<a
href="http://arxiv.org/abs/1410.1219"
target="_blank"
rel="noopener noreferrer"
>2015</a
>] disproved their conjecture, showing that for odd \(t\) with
\((t+1)(t+2) \leq r\), there exists an \(r\)-regular graph with no
\(\{r-t,t\}\)-factor.
<br />
The smallest values of \(r\) and \(t\) for which Akbari and Kano's
conjecture remains open are \(r = 5\) and \(t = 1\).
<br />
<b>Conjecture:</b> Every \(5\)-regular pseudograph contains a
\(\{4,1\}\)-factor.
<br />
Bernshteyn et al. [<a
href="http://arxiv.org/abs/1603.09384"
target="_blank"
rel="noopener noreferrer"
>2016+</a
>] established about a dozen conditions that must apply to a
vertex-minimal \(5\)-regular pseudograph with no \(\{4,1\}\)-factor,
provided such a graph exists. <br /><br />
</li>
</ul>
<b>... in Geometry</b>
<ul>
<li>
<b>Question:</b> What is the greatest number of non-overlapping
congruent regular tetrahedra that can share a common vertex?
<br />
You can slice up a regular icosahedron into \(20\) non-regular
tetrahedra touching the center, and these are short, fat tetrahedra,
so you can fit a regular tetrahedron within each.
<br />
By calculating the solid angle cut out by a regular tetrahedron, you
find that at most \(22\) can fit within the full solid angle.
<br />
This question first emerged no later than 1958, yet it remains open
whether the answer is \(20\), \(21\), or \(22\).
<br />
Lagarias and Zong [<a
href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.251.3224"
target="_blank"
rel="noopener noreferrer"
>2012</a
>] give more details for this and other tetrahedra-packing problems.
<br /><br />
</li>
<li>
There are many ways one might define how "close" a simple polygon is
to being convex. For example, every exterior angle of a polygon has
measure at least \(\pi\), so a polygon with a minimum exterior angle
near zero might be considered "very unconvex."
<br />
<b>Conjecture 1:</b> Every set of \(n\) points (no \(3\) colinear) has
a simple polygonization with minimum exterior angle at least
\(\frac{2\pi}{n-1}\).
<br />
Rorabaugh [<a
href="http://arxiv.org/abs/1409.4344"
target="_blank"
rel="noopener noreferrer"
>2014+</a
>] posed this along with the following partial results:
<br />
(i) ... minimum exterior angle at least \(\frac{\pi}{(n-1)(n-x)}\),
where \(x\) is the number of points on the convex hull.
<br />
(ii) If the conjecture is correct, it is tight, achieved by \(n-1\)
points in a circle with \(1\) point in the center.
<br /><br />
</li>
<li>
A subset \(S \subseteq R^d\) is a <i>Danzer set</i> provided every
convex body of volume \(1\) in \(R^d\) contains a point in \(S\).
Assume \(d \geq 2\).
<br />
<b>Question:</b> Does there exist a Danzer set with density
\(O(1)\)--that is, with \(O(r^d)\) points in the ball of radius \(r
\rightarrow \infty\)?
<br />
Bambah and Woods [<a
href="http://projecteuclid.org/download/pdf_1/euclid.pjm/1102970604"
target="_blank"
rel="noopener noreferrer"
>1971</a
>] showed that a Danzer set cannot be the union of a finite number of
grids, but there exist Danzer sets with density \(O\!\left((\log
r)^{(d-1)}\right)\).
<br />
Solomon and Weiss [<a
href="http://arxiv.org/abs/1406.3807"
target="_blank"
rel="noopener noreferrer"
>2014</a
>] improved both results, showing that a Danzer set cannot be a
cut-and-project set, but there exist Danzer sets with density \(O(\log
r)\). <br /><br />
Gowers [<a
href="http://www.dpmms.cam.ac.uk/~wtg10/gafavisions.ps"
target="_blank"
rel="noopener noreferrer"
>2000</a
>] asked whether there exists a Danzer set \(S\) with \(d = 2\) and a
constant \(C\) such that every convex body of area \(1\) contains at
most \(C\) points in \(S\). <br />
Solan, Solomon, and Weiss [<a
href="http://arxiv.org/abs/1510.07179"
target="_blank"
rel="noopener noreferrer"
>2015</a
>] proved that no such set exists; moreover, given a Danzer set \(S\)
with \(d \geq 2\), for any positve \(n\) and \(\varepsilon\), there
exists an ellipsoid with volume at most \(\varepsilon\) and with at
least \(n\) points in \(S\). <br /><br />
Conway offers
<a
href="http://oeis.org/A248380/a248380.pdf"
target="_blank"
rel="noopener noreferrer"
>$1000</a
>
for a solution to the following Danzer problem variant:
<br />
"<b>Dead Fly Problem:</b> If a set of points in the plane contains one
point in each convex region of area \(1\), then must it have pairs of
points at arbitrarily small distances?"
<br />
The Solan-Solomon-Weiss result does not give a positive answer to
Conway's question because the longest diameter of their ellipsoid can
grow with \(n\).
<br /><br />
</li>
<li>
A point-set \(S\) in the plane satisfies
<i>geometric characterization \(GC_n\)</i> provided for each \(x \in
S\), there is a set of \(n\) lines such that \(x\) is not on any of
the lines by each point in \(S-\{x\}\) is on at least one line.
<br />
<b>Conjecture:</b> Given a \(GC_n\) point-set \(S\), there exists a
line containing \(n-1\) points of \(S\).
<br />
This is a conjecture of Gasca and Maeztu [<a
href="http://resolver.sub.uni-goettingen.de/purl?GDZPPN001177109"
target="_blank"
rel="noopener noreferrer"
>1982</a
>].
<br />
It is known to be true for \(n \leq 5\): Busch [<a
href="http://www.researchgate.net/publication/268997837_A_note_on_Lagrange_interpolation_in_R_2"
target="_blank"
rel="noopener noreferrer"
>1990</a
>] proved it for \(n = 4\) and Hakopian, Jetter, and Zimmermann [<a
href="http://www.uni-hohenheim.de/~gzim/Publications/gmcfnf.pdf"
target="_blank"
rel="noopener noreferrer"
>2013</a
>] for \(n = 5\).
<br />
This problem was my
<a
href="http://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxybWdwZ3J3Y3xneDoyNDc4ZTdjNWQ0ZDJlN2M5"
target="_blank"
rel="noopener noreferrer"
>2014</a
>
submission to the Rocky Mountain-Great Plains Graduate Research
Workshop in Combinatorics. <br /><br />
</li>
<li>
Call two filled triangle in \(\mathbb{R}^3\)
<i>almost-disjoint</i> provided they intersect in at most one point
and, if they do, that point is a vertex of both triangles. Let
\(f(n)\) be the maximum number of pairwise almost-disjoint triangles
one can embed in \(\mathbb{R}^3\) so that the total number of vertex
points is \(n\). For example \(4\) faces of an octahedron can be
selected so that no two share more than one point, and this is the
greatest number of pairwise almost-disjoint triangles on \(6\) vertex
points, of so \(f(6) = 4\).
<br />
<b>Question:</b> What is the asymptotic growth of \(f(n)\)?
<br />
Since each pair of vertices is contained in at most one edge, and
there are only three edges per triangle, \(f(n) \leq {n \choose 2}/3
\ll n^2\).
<br />
This question was asked by Kalai in 1995, as told by Károlyi
and Solymosy [<a
href="https://www.semanticscholar.org/paper/Almost-Disjoint-Triangles-in-3-Space-K%C3%A1rolyi-Solymosi/b1b5eb7576416aeb5d9aaf8ed99ca21eb129257e"
target="_blank"
rel="noopener noreferrer"
>2002</a
>], who proved that \(f(n) \gg n^{3/2}\).
<br />
This problem was my
<a
href="https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxybWdwZ3J3Y3xneDozN2Y1Y2FjZjA3NWExOGMx"
target="_blank"
rel="noopener noreferrer"
>2015</a
>
submission to the Rocky Mountain-Great Plains Graduate Research
Workshop in Combinatorics. <br /><br />
</li>
<li>
An <i>intersecting family</i> is a collection of sets such that every
pair of sets has a non-empty intersection. An \(r\)-set is a set with
\(r\) elements. <br />
Erdős, Ko, and Rado [<a
href="http://www.renyi.hu/~p_erdos/1961-07.pdf"
target="_blank"
rel="noopener noreferrer"
>1961</a
>] proved that the maximum size of an intersecting family of
\(r\)-subsets of \(\{1,2,...,n\}\) with \(2r \leq n\) is \({n-1
\choose r-1}\). <br />
Equality can be attained by fixing one element and taking the
collection of all \(r\)-sets containing that element. Thus there are
\(n-1\) options for the other \(r-1\) elements. <br />
Kalai [<a
href="http://mathoverflow.net/questions/114646"
target="_blank"
rel="noopener noreferrer"
>2015</a
>] proposed an analogous result for polygon triangulations: <br />
"Let \(\mathcal{T}_n\) be the family of all triangulations on an
\(n\)-gon using \(n-3\) non-intersecting diagonals. The number of
triangulations in \(\mathcal{T}_n\) is \(C_{n-2}\) the \((n-2)\)-th
<a
href="http://oeis.org/A000108"
target="_blank"
rel="noopener noreferrer"
>Catalan number</a
>. Let \(\mathcal{S} \subset \mathcal{T}_n\) be a subfamily of
triangulations with the property that every two triangulations of
\(\mathcal{S}\) have a common diagonal.<br />
<b>Problem:</b> Show that \(|\mathcal{S}| \leq
|\mathcal{T}_{n-1}|\)."<br />
Similarly to the set situation, equality can be attained by fixing a
diagonal that divides the n-gon into a triangle and an \((n-1\))-gon,
leaving \(|\mathcal{T}_{n-1}|\) possibilities for the rest of the
triangulation.
<br /><br />
</li>
</ul>
<b>... on Words</b>
<ul>
<!--
<li>
Word W is an <i>instance</i> of word V provided W = φ(V) for some nonerasing monoid homomorphism φ.
For example, "Mr.Freeze" is an instance of "cool" via the homomorphism defined by φ(c) = Mr.Fr, φ(o) = e, φ(l) = ze.
The <i>instance probability</i> I_n(V,q) is defined to be the number of length-n instances of V over the alphabet {1,2,...,q} divided by q^n.
For V consisting of k>0 distinct letters, let r_1,...,r_k be the number of times each letter appears in V, and let d = gcd(r_i) and r = min(r_i).
<br>
<b>Conjecture 13:</b> lim_{n->∞, d|n} I_n(V,q)*q^(n*(1 - 1/r)) exists.
<br>
Cooper and Rorabaugh [<a href=" in <a href="http://arxiv.org/abs/1504.04424" target="_blank" rel="noopener noreferrer">2015</a>] posed this along with the following partial results:
<br>
(i) When r = 1, the limit exists and is positive.
<br>
(ii) When r divides |V|, the limit (provided it exists) must be at least q^(k - |V|/r).
<br>
(iii) When d divides n (otherwise the instance probability is 0), I_n(V,q)*q^(n*(1 - 1/r)) is bounded above by binomial(n/d + k + 1, k + 1).
<br><br>
-->
<li>
The <i>period</i> of word \(W\), denoted \(p(W)\), is the least
positive \(p\) such that \(W[i] = W[i+p]\) for all \(i\). A
<i>bifix</i> or <i>border</i> of \(W\) is a proper factor that is both
a prefix and suffix of \(W\). Thus \(p(W) = |W|\) iff \(W\) is
bifix-free/unbordered. Let \(\mu(W)\) be the length of the largest
bifix-free factor of \(W\). Trivially, \(\mu(W) \leq p(W)\).
<br />
<b>Question:</b> What word-lengths \(|W|\) imply that \(\mu(W) =
p(W)\)?
<br />
This was first explored by Ehrenfeucht and Silberger [<a
href="http://dx.doi.org/10.1016/0012-365X(79)90116-X"
target="_blank"
rel="noopener noreferrer"
>1979</a
>], who conjectured that \(|W| \geq 2\mu(W)\) is sufficient.
<br />
Assous and Pouzet [<a
href="http://dx.doi.org/10.1016/0012-365X(79)90146-8"
target="_blank"
rel="noopener noreferrer"
>1979</a
>] disproved that conjecture, providing couterexamples when \(|W| =
\frac{7}{3}\mu(W) - 4\). They conjectured that \(|W| \geq 3\mu(W)\) is
sufficient and best possible.
<br />
Duval [<a
href="http://dx.doi.org/10.1016/0012-365X(82)90186-8"
target="_blank"
rel="noopener noreferrer"
>1982</a
>] proved that \(|W| \geq 4\mu(W) - 6\) is sufficient.
<br />
Harju and Nowotka [<a
href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.101.4863&rep=rep1&type=pdf"
target="_blank"
rel="noopener noreferrer"
>2003</a
>] proved that \(|W| \geq 3\mu(W) - 2\) is sufficient and conjectured
that even \(|W| \geq \frac{7}{3}\mu(W) - 3\) suffices. <br /><br />
</li>
<li>
<b>Question:</b> Do there exist words \(u_1, \ldots, u_n\) such that
<ul>
<li>
\((u_1 u_2 \cdots u_n)^2 = (u_1)^2 (u_2)^2 \cdots (u_n)^2\) and
</li>
<li>
\((u_1 u_2 \cdots u_n)^3 = (u_1)^3 (u_2)^3 \cdots (u_n)^3\),
</li>
</ul>
and the words do not all pairwise commute, that is, \(u_i u_j \neq u_j
u_i\) for at least one pair \((i,j)\)?<br />
Holub [<a
href="http://msekce.karlin.mff.cuni.cz/~holub/soubory/prizeproblem.pdf"
target="_blank"
rel="noopener noreferrer"
>2003+</a
>] offers a reward of 200 € for a solution. <br /><br />
</li>
</ul>
<b>... on Numbers</b>
<ul>
<li>
The <b>Collatz Conjecture</b> or <b>\(3n+1\) Problem</b>.
<br />
Consider the following process: if a number is odd, multiply by three
and add one; if it is even, divide by two.
<br />
<b>Conjecture:</b> Every positive integer, under repeated application
of the \(3n+1\) process, will eventually reach \(1\).
<br />
Starting with \(7\), for example, \(7 \rightarrow 22 \rightarrow 11
\rightarrow 34 \rightarrow 17 \rightarrow 52 \rightarrow 26
\rightarrow 13 \rightarrow 40 \rightarrow 20 \rightarrow 10
\rightarrow 5 \rightarrow 16 \rightarrow 8 \rightarrow 4 \rightarrow 2
\rightarrow 1\).
<br />
You can read more on
<a
href="http://mathworld.wolfram.com/CollatzProblem.html"
target="_blank"
rel="noopener noreferrer"
>MathWorld</a
>
or
<a
href="http://en.wikipedia.org/wiki/Collatz_conjecture"
target="_blank"
rel="noopener noreferrer"
>Wikipedia</a
>.
<br />
This is an extremely popular problem, as demonstrated by Lagarias'
annotated bibliographies (<a
href="http://arxiv.org/abs/math/0309224"
target="_blank"
rel="noopener noreferrer"
>1963—1999</a
>
and
<a
href="http://arxiv.org/abs/math/0608208"
target="_blank"
rel="noopener noreferrer"
>2000—2009</a
>) and the hundreds of
<a
href="http://oeis.org/search?q=collatz"
target="_blank"
rel="noopener noreferrer"
>"collatz"</a
>-related sequences on the OEIS.
<br />
I myself investigated a generalization for my undergraduate thesis [<a
href="docs/CollatzGeneralized.pdf"
target="_blank"
rel="noopener noreferrer"
>2010</a
>], which I have since enjoyed sharing with other undergrads [<a
href="http://www.youtube.com/watch?v=mYX9PI6dAzs"
target="_blank"
rel="noopener noreferrer"
>2012</a
>].
<br />
As of March 2016, the conjecture has been verified for every positive
integer up to \(2^{60}\), according to Roosendaal's
<a
href="http://www.ericr.nl/wondrous/"
target="_blank"
rel="noopener noreferrer"
>website</a
>.
<br />
Before you descend into this rabbit hole, consider a
<a
href="http://xkcd.com/710/"
target="_blank"
rel="noopener noreferrer"
>warning</a
>
from xkcd. <br /><br />
</li>
</ul>
</div>
<hr />
<br />
Please email me if you are aware of any significant advances on the above
problems or if any of my hyperlinks are dead.
<br /><br />
<hr />
<!--Other Lists-->
<div>
<h4>Other Problem Sources</h4>
<ul>
<li>
Clark Kimberling,
<a
href="http://faculty.evansville.edu/ck6/integer/unsolved.html"
target="_blank"
rel="noopener noreferrer"
>Unsolved Problems and Rewards</a
>
(sequences)
</li>
<li>
Craig Larson and Nico Van Cleemput,
<a
href="http://independencenumber.wordpress.com/"
target="_blank"
rel="noopener noreferrer"
>The Conjecturing Project</a
>
(graphs)
</li>
<li>
Dan Archdeacon,
<a
href="http://www.cems.uvm.edu/TopologicalGraphTheoryProblems/"
target="_blank"
rel="noopener noreferrer"
>Problems in Topological Graph Theory</a
>
</li>
<li>
David Eppstein,
<a
href="http://www.ics.uci.edu/%7Eeppstein/junkyard/open.html"
target="_blank"
rel="noopener noreferrer"
>The Geometry Junkyard: Open Problems</a
>
(discrete and computational geometry)
</li>
<li>
Douglas B. West,
<a
href="http://www.math.illinois.edu/~dwest/openp/"
target="_blank"
rel="noopener noreferrer"
>Open Problems - Graph Theory and Combinatorics</a
>
</li>
<li>
Douglas B. West,
<a
href="http://www.math.illinois.edu/~dwest/regs/"
target="_blank"
rel="noopener noreferrer"
>REGS in Combinatorics - Univ. of Illinois</a
>
</li>
<li>
Egerváry Research Group,
<a
href="http://lemon.cs.elte.hu/egres/open"
target="_blank"
rel="noopener noreferrer"
>Egres Open</a
>
(combinatorial optimization)
</li>
<li>
Erik D. Demaine, Joseph S. B. Mitchell, and Joseph O'Rourke,
<a
href="http://cs.smith.edu/~orourke/TOPP"
target="_blank"
rel="noopener noreferrer"
>The Open Problems Project</a
>
(computational geometry and related fields)
</li>
<li>
Jerry Spinrad,
<a
href="http://www.vuse.vanderbilt.edu/%7Espin/open.html"
target="_blank"
rel="noopener noreferrer"
>open</a
>
(graphs, graph algorithms)
</li>
<li>
Joshua Cooper,
<a
href="http://people.math.sc.edu/cooper/combprob.html"
target="_blank"
rel="noopener noreferrer"
>Combinatorial Problem I Like</a
>
</li>
<li>
MathOverflow,
<a
href="http://mathoverflow.net/search?q=answers%3A0+closed%3Ano"
target="_blank"
rel="noopener noreferrer"
>Unanswered Questions</a
>
</li>
<li>
The On-line Encyclopedia of Integer Sequence,
<a
href="https://oeis.org/search?q=conjecture+-abc+-collatz+-goldbach+-twin"
target="_blank"
rel="noopener noreferrer"
>sequences with "conjecture" in their entry</a
>
</li>
<li>
The On-line Encyclopedia of Integer Sequence,
<a
href="https://oeis.org/search?q=empirical"
target="_blank"
rel="noopener noreferrer"
>sequences with "empirical" in their entry</a
>
</li>
<li>
<a
href="http://www.openproblemgarden.org/"
target="_blank"
rel="noopener noreferrer"
>Open Problem Garden</a
>
</li>
<li>
Peter J. Cameron,
<a
href="http://www.maths.qmul.ac.uk/~pjc/oldprobidx.html"
target="_blank"
rel="noopener noreferrer"
>Problem pages index</a
>
(combinatorics)
</li>
<li>
Rocky Mountain-Great Plains Graduate Research Workshop in
Combinatorics,
<a
href="http://sites.google.com/site/rmgpgrwc/problem-gardens"
target="_blank"
rel="noopener noreferrer"
>Problem Gardens</a
>
</li>
<li>
Rutgers Center for Discrete Mathematics & Theoretical Computer
Science,
<a
href="http://dimacs.rutgers.edu/Challenges/"
target="_blank"
rel="noopener noreferrer"
>DIMACS Implementation Challenges</a
>
</li>
<li>
S.C. Locke,
<a
href="http://math.fau.edu/locke/Unsolved.htm"
target="_blank"
rel="noopener noreferrer"
>Unsolved Problems</a
>
(graphs, etc.)
</li>
<li>
UCSD Mathematics Department,
<a
href="http://www.math.ucsd.edu/~erdosproblems"
target="_blank"
rel="noopener noreferrer"
>Erdős' Problems on Graphs</a
>
</li>
<li>
University of Waterloo,
<a
href="http://cs.uwaterloo.ca/twiki/view/CoWiki/WebHome"
target="_blank"
rel="noopener noreferrer"
>CoWiki</a
>
(words)
</li>
<li>
Vašek Chvátal,
<a
href="http://users.encs.concordia.ca/%7Echvatal/perfect/problems.html"
target="_blank"
rel="noopener noreferrer"
>Perfect Problems</a
>
(perfect graphs)
</li>
<li>
Wikipedia,
<a
href="http://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics"
target="_blank"
rel="noopener noreferrer"
>List of unsolved problems in mathematics</a
>
</li>
<li>
Wolfram MathWorld,
<a
href="http://mathworld.wolfram.com/UnsolvedProblems.html"
target="_blank"
rel="noopener noreferrer"
>Unsolved Problems</a
>
</li>
</ul>
</div>
<br />
</body>
</html>