Preamble ...................................................... vii
Preliminary remarks ............................................ ix
Introduction .................................................... 1
1. General notions and motivations .............................. 1
2. Organization and presentation of the results ................. 6
Chapter 1. FraÏssė classes of finite metric spaces and
Urysohn spaces ...................................... 13
1. Fundamentals of FraÏssė theory ............................. 13
2. Amalgamation and FraÏssė classes of finite metric spaces ... 17
3. Urysohn spaces .............................................. 27
4. Complete separable ultrahomogeneous metric spaces ........... 32
Chapter 2. Ramsey calculus, Ramsey degrees and universal
minimal flows ....................................... 37
1. Fundamentals of Ramsey theory and topological dynamics ...... 37
2. Finite metric Ramsey theorems ............................... 40
3. Ordering properties ......................................... 55
4. Ramsey degrees .............................................. 60
5. Universal minimal flows and extreme amenability ............. 61
6. Concluding remarks and open problems ........................ 69
Chapter 3. Big Ramsey degrees, indivisibility and
oscillation stability ............................... 73
1. Fundamentals of infinite metric Ramsey calculus and
oscillation stability ....................................... 73
2. Big Ramsey degrees .......................................... 76
3. Indivisibility .............................................. 77
4. Approximate indivisibility and oscillation stability ....... 102
5. Concluding remarks and open problems ....................... 113
Appendix A. Amalgamation classes Ms when |S| ≤ 4 .............. 115
6. |S|=3 ...................................................... 115
7. |S|=4 ...................................................... 116
Appendix B. Indivisibility of Us when |S| ≤ 4 ................. 125
Appendix C. On the universal Urysohn space U .................. 129
Bibliography .................................................. 133
Index ......................................................... 137
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