Formal Book

1 Six proofs of the infinity of primes ▶
- 1.1 Appendix: Infinitely many more proofs
2 Bertrand’s postulate ▶
- 2.1 Appendix: Some estimates
3 Binomial coefficients are (almost) never powers
4 Representing numbers as sums of two squares
5 The law of quadratic reciprocity
6 Every finite division ring is a field
7 The spectral theorem and Hadamard’s determinant problem
8 Some irrational numbers
9 Four times \(π^2/6\)
10 Hilbert’s third problem: decomposing polyhedra
11 Lines in the plane and decompositions of graphs
12 The slope problem
13 Three applications of Euler’s formula
14 Cauchy’s rigidity theorem
15 The Borromean rings don’t exist
16 Touching simplices
17 Every large point set has an obtuse angle
18 Borsuk’s conjecture
19 Sets, functions, and the continuum hypothesis ▶
- Appendix: On cardinal and ordinal numbers
20 In praise of inequalities
21 The fundamental theorem of algebra
22 One square and an odd number of triangles ▶
- Appendix: Extending valuations
23 A theorem of Pólya on polynomials ▶
- 23.1 Appendix: Chebyshev’s theorem
24 Van der Waerden’s permanent conjecture
25 On a lemma of Littlewood and Offord
26 Cotangent and the Herglotz trick
27 Buffon’s needle problem
28 Pigeon-hole and double counting ▶
29 Tiling rectangles
30 Three famous theorems on finite sets
31 Shuffling cards
32 Lattice paths and determinants
33 Cayley’s formula for the number of trees
34 Identities versus bijections
35 The finite Kakeya problem
36 Completing Latin squares
37 Permanents and the power of entropy ▶
- 37.1 Appendix: More about entropy
38 The Dinitz problem
39 Five-coloring plane graphs
40 How to guard a museum
41 Turán’s graph theorem
42 Communicating without errors
43 The chromatic number of Kneser graphs ▶
- 43.1 Appendix: A proof sketch for the Borsuk–Ulam theorem
44 Of friends and politicians
45 Probability makes counting (sometimes) easy
Dependency graph

39 Five-coloring plane graphs

Theorem 39.1

#

All planar graphs \(G\) can be \(5\)-colored:

\[ \chi _{\ell }(G)\le 5. \]

Proof ▶

TODO