Presentazione Tesi Magistrale

1. Gowers’ Ramsey Theorem Pietro Porqueddu University of Pisa 03/02/2017 Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 1 / 37

2. Introduction In 1992, Timothy Gowers provided a positive answer to the oscillation stability problem in c0. The very core of the proof is a combinatorial argument which represents a very important result in Ramsey Theory known as Gowers’ Ramsey Theorem. One of the typical problems is Ramsey Theory is to determine whether some structure is preserved when it is partitioned (coloured). Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 2 / 37

3. The distortion problem Let (E, · ) be a Banach space and λ > 1 a real number. We say that E is λ-distortable if there exists an equivalent norm | · | on E such that for every inﬁnite-dimensional vector subspace X of E, sup |x| |y| | x, y ∈ X and ||x|| = ||y|| = 1 ≥ λ. We say that E is distortable if it is λ-distortable for some λ > 1, and it is arbitrarily distortable if it is λ-distortable for every λ > 1. The question of whether a Banach space is distortable or not is called the distortion problem. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 3 / 37

4. The distortion problem Let (E, · ) be a Banach space and λ > 1 a real number. We say that E is λ-distortable if there exists an equivalent norm | · | on E such that for every inﬁnite-dimensional vector subspace X of E, sup |x| |y| | x, y ∈ X and ||x|| = ||y|| = 1 ≥ λ. We say that E is distortable if it is λ-distortable for some λ > 1, and it is arbitrarily distortable if it is λ-distortable for every λ > 1. The question of whether a Banach space is distortable or not is called the distortion problem. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 3 / 37

5. The distortion problem R.C. James (1964) proved that c0 and 1 are not distortable. V. Milman (1971) proved that a non-distortable space must contain an isomorphic copy of c0 or p, 1 ≤ p < ∞. The problem in the case of separable Hilbert spaces and for p, for any 1 < p < ∞, was solved aﬃrmatively by E. Odell and T. Schlumprecht (1994). Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 4 / 37

8. The oscillation stability problem In a Banach space (E, · ), the oscillation stability problem is the question of whether or not, given > 0, for every real-valued Lipschitz function f on the unit sphere SE there exists an inﬁnite-dimensional subspace X of E on the unit sphere of which f varies by at almost , i.e., whether there is a real number a ∈ R and an inﬁnite-dimensional subspace X of E such that ||a − f (x)|| < for all x ∈ X with ||x|| = 1. In a separable and uniform convex Banach space (such as the p spaces for 1 < p < ∞), the distortion problem and the oscillation stability problem are equivalent. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 5 / 37

9. The oscillation stability problem In a Banach space (E, · ), the oscillation stability problem is the question of whether or not, given > 0, for every real-valued Lipschitz function f on the unit sphere SE there exists an inﬁnite-dimensional subspace X of E on the unit sphere of which f varies by at almost , i.e., whether there is a real number a ∈ R and an inﬁnite-dimensional subspace X of E such that ||a − f (x)|| < for all x ∈ X with ||x|| = 1. In a separable and uniform convex Banach space (such as the p spaces for 1 < p < ∞), the distortion problem and the oscillation stability problem are equivalent. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 5 / 37

10. Gowers’ Theorem Theorem Let > 0 and let F be a real-valued Lipschitz function on the unit sphere of c0. Then there is an inﬁnite-dimensional subspace of c0 on the unit sphere of which F varies by at most . The strategy of the proof is to ﬁnd explicit generators of such a subspace taken among the maps which belong to a δ-net on Sc0 . Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 6 / 37

11. Gowers’ Theorem Theorem Let > 0 and let F be a real-valued Lipschitz function on the unit sphere of c0. Then there is an inﬁnite-dimensional subspace of c0 on the unit sphere of which F varies by at most . The strategy of the proof is to ﬁnd explicit generators of such a subspace taken among the maps which belong to a δ-net on Sc0 . Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 6 / 37

12. What we will see • In the ﬁrst part, we will introduce the notions of ultraﬁlter and semigroup, and some of their properties. • In the second part, we will prove Gowers’ Ramsey Theorem. • Finally, we will see Gowers’ solution of the oscillation stability problem in c0 as a consequence of Gowers’ Ramsey Theorem. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 7 / 37

13. Semigroups Definition A semigroup is a nonempty set S with a map · : S2 −→ S defined for all x, y ∈ S, that satisfies the associative law (x · y) · z = x · (y · z) Usually we will drop the · and we will write xy instead of x · y. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 8 / 37

14. Ideals and subsemigroups Deﬁnition A compact left-semigroup is a nonempty semigroup S with a compact Hausdorﬀ topology such that, for all x ∈ S, the map λx : y −→ xy is continuous for all y. A subset T ⊆ S is a (compact) subsemigroup of S if it is a compact left-semigroup as subspace of S. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 9 / 37

15. Idempotents and Ellis’ Theorem Deﬁnition An element x ∈ S is idempotent if and only if x2 = x. Theorem (Ellis’ Theorem) Every compact left-semigroup S has an idempotent. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 10 / 37

16. Idempotents and Ellis’ Theorem Deﬁnition An element x ∈ S is idempotent if and only if x2 = x. Theorem (Ellis’ Theorem) Every compact left-semigroup S has an idempotent. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 10 / 37

17. Filters and ultrafilters Definition Let S be a set. A family F of subsets of S is a filter on S if 1 ∅ /∈ F and S ∈ F; 2 If A, B ∈ F, then A ∩ B ∈ F; 3 If A ∈ F and A ⊆ B, then B ∈ F. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 11 / 37

18. Filters and ultrafilters Proposition Let S be a set and F a filter on S. The following are equivalent 1 If A /∈ F, then Ac ∈ F; 2 If A1 ∪ . . . ∪ An ∈ F, then there exists i such that Ai ∈ F; 3 F is maximal respect to the inclusion, i.e. if G is a filter on S and F ⊆ G, then F = G. Definition A filter F on a set S which satisfies one, and then all, of the properties of the above proposition is called ultrafilter. We will denote by βS the set of all ultrafilters on S. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 12 / 37

21. The space βS Definition On βS we define the topology generated by the (cl)open set basis B = {OA | A ⊆ S} where OA = {U ultrafilter on S | A ∈ U} Theorem The space βS is the Stone-ˇCech compactification of the discrete space S: 1 βS is a compact Hausdorff space; 2 S is a dense discrete subset of βS; 3 for any compact Hausdorff space K and any f : S → K, there exists an unique continuous extension ¯f : βS → K. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 13 / 37

22. The space βS Definition On βS we define the topology generated by the (cl)open set basis B = {OA | A ⊆ S} where OA = {U ultrafilter on S | A ∈ U} Theorem The space βS is the Stone-ˇCech compactification of the discrete space S: 1 βS is a compact Hausdorff space; 2 S is a dense discrete subset of βS; 3 for any compact Hausdorff space K and any f : S → K, there exists an unique continuous extension ¯f : βS → K. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 13 / 37

23. The space βS Theorem Let (S, ·) be a compact semigroup. There exists a unique associative binary operation ∗ : βS × βS → βS satisfying the following conditions 1 For every x, y ∈ S, x ∗ y = x · y; 2 For each V ∈ βS, the function pV : βS × βS → βS is continuous, where pV(U) = U ∗ V; 3 For each x ∈ S, the function qx : βS × βS → βS is continuous, where qx (V) = x ∗ V. Proposition Let U, V be ultraﬁlters in βS. Then A ∈ U ∗ V ⇔ {x ∈ S | {y ∈ S | x · y ∈ A} ∈ V} ∈ U. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 14 / 37

24. The space βS Theorem Let (S, ·) be a compact semigroup. There exists a unique associative binary operation ∗ : βS × βS → βS satisfying the following conditions 1 For every x, y ∈ S, x ∗ y = x · y; 2 For each V ∈ βS, the function pV : βS × βS → βS is continuous, where pV(U) = U ∗ V; 3 For each x ∈ S, the function qx : βS × βS → βS is continuous, where qx (V) = x ∗ V. Proposition Let U, V be ultraﬁlters in βS. Then A ∈ U ∗ V ⇔ {x ∈ S | {y ∈ S | x · y ∈ A} ∈ V} ∈ U. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 14 / 37

25. Algebra in βS Proposition The ultraﬁlter product U ∗ V has the following properties 1 (U ∗ V) ∗ W = U ∗ (V ∗ W); 2 V → U ∗ V is a continuous map from βS into βS for every U. Corollary The space (βS, ∗) is a compact left-semigroup. Corollary There exist idempotent ultraﬁlters in (βS, ∗). Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 15 / 37

28. The set FIN± k Let k be a positive integer. We consider the set FIN± k of the maps p : N → {0, ±1, . . . , ±k} such that supp(p) = {n ∈ N | p(n) = 0} is ﬁnite, and p attains at least one of the values k or −k. We deﬁne on FIN± k a coordinate-wise sum (p + q)(n) = p(n) + q(n) whenever supp(p) ∩ supp(q) = ∅. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 16 / 37

31. Ultrafilters on FIN± k The set FIN± k is an example of partial semigroup. The construction of a semigroup of ultrafilters (βS, ∗) works also for a large class of partial semigroups S. In this case, we need to restrict to cofinite ultrafilters. Definition We say that an ultrafilter U on FIN± k is cofinite if ∀p ∈ FIN± k , {q ∈ FIN± k | p + q is defined} ∈ U. We denote by γFIN± k the set of all cofinite ultrafilters. Proposition The set (γFIN± k , +) is a compact left-semigroup. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 17 / 37

32. Ultrafilters on FIN± k The set FIN± k is an example of partial semigroup. The construction of a semigroup of ultrafilters (βS, ∗) works also for a large class of partial semigroups S. In this case, we need to restrict to cofinite ultrafilters. Definition We say that an ultrafilter U on FIN± k is cofinite if ∀p ∈ FIN± k , {q ∈ FIN± k | p + q is defined} ∈ U. We denote by γFIN± k the set of all cofinite ultrafilters. Proposition The set (γFIN± k , +) is a compact left-semigroup. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 17 / 37

33. Ultraﬁlters on FIN± k We extend the sum to the partial semigroup FIN± [1,k] = k i=1 FIN± i , If p ∈ FIN± k and q ∈ FIN± h , then p + q ∈ FIN± max{k,h}. The correspondent space of ultraﬁlters is γ(FIN± [1,k]) = k i=1 γFIN± i Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 18 / 37

34. Ultraﬁlters on FIN± k We extend the sum to the partial semigroup FIN± [1,k] = k i=1 FIN± i , If p ∈ FIN± k and q ∈ FIN± h , then p + q ∈ FIN± max{k,h}. The correspondent space of ultraﬁlters is γ(FIN± [1,k]) = k i=1 γFIN± i Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 18 / 37

35. The tetris operation Deﬁnition We call tetris operation the map T : FIN± k → FIN± k−1 deﬁned as T(p)(n) =    p(n) − 1 if p(n) > 0, 0 if p(n) = 0, p(n) + 1 if p(n) < 0. Notice that T(p + q) = T(p) + T(q) whenever p, q have disjoint supports. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 19 / 37

36. The tetris operation Deﬁnition We call tetris operation the map T : FIN± k → FIN± k−1 deﬁned as T(p)(n) =    p(n) − 1 if p(n) > 0, 0 if p(n) = 0, p(n) + 1 if p(n) < 0. Notice that T(p + q) = T(p) + T(q) whenever p, q have disjoint supports. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 19 / 37

37. Block sequences Deﬁnition A basic block sequence B = {bn}n of elements of FIN± k is any sequence such that supp(bi ) < supp(bj ) whenever i < j. Deﬁnition The partial subsemigroup of FIN± k generated by a basic block sequence B = {bn}n is the family of functions of the form 0Tj0 (bn0 ) + 1Tj1 (bn1 ) + . . . + l Tjl (bnl ), where i = ±1, n0 < . . . < nl , j0, . . . , jl ∈ {0, . . . , k − 1} and at least one of the j0, . . . , jl is equal to zero. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 20 / 37

38. Block sequences Deﬁnition A basic block sequence B = {bn}n of elements of FIN± k is any sequence such that supp(bi ) < supp(bj ) whenever i < j. Deﬁnition The partial subsemigroup of FIN± k generated by a basic block sequence B = {bn}n is the family of functions of the form 0Tj0 (bn0 ) + 1Tj1 (bn1 ) + . . . + l Tjl (bnl ), where i = ±1, n0 < . . . < nl , j0, . . . , jl ∈ {0, . . . , k − 1} and at least one of the j0, . . . , jl is equal to zero. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 20 / 37

39. Extending the tetris operation We extend the tetris operation on the space of coﬁnite ultraﬁlters as follows: T : γFIN± k → γFIN± k−1 is the function where T(U) = {A ⊆ FIN± k−1 | {p ∈ FIN± k |T(p) ∈ A} ∈ U}. for all U ∈ γFIN± k . Proposition T : γFIN± k → γFIN± k−1 is a continuous onto homomorphism. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 21 / 37

40. Extending the tetris operation We extend the tetris operation on the space of coﬁnite ultraﬁlters as follows: T : γFIN± k → γFIN± k−1 is the function where T(U) = {A ⊆ FIN± k−1 | {p ∈ FIN± k |T(p) ∈ A} ∈ U}. for all U ∈ γFIN± k . Proposition T : γFIN± k → γFIN± k−1 is a continuous onto homomorphism. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 21 / 37

41. Subsymmetric ultrafilters For every A ⊆ FIN± k and > 0, we define (A) = {q ∈ FIN± k | ∃p ∈ A p − q ∞≤ } Definition We say that a cofinite ultrafilter U ∈ γFIN± k is subsymmetric if −(A)1 ∈ U for all A ∈ U. We denote by S± k the set of all subsymmetric ultrafilters on FIN± k . Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 22 / 37

42. Subsymmetric ultrafilters For every A ⊆ FIN± k and > 0, we define (A) = {q ∈ FIN± k | ∃p ∈ A p − q ∞≤ } Definition We say that a cofinite ultrafilter U ∈ γFIN± k is subsymmetric if −(A)1 ∈ U for all A ∈ U. We denote by S± k the set of all subsymmetric ultrafilters on FIN± k . Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 22 / 37

43. Subsymmetric ultraﬁlters Lemma The set S± k is a closed subsemigroup of γFIN± k . Lemma T(U) ∈ S± k−1 for all U ∈ S± k . Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 23 / 37

44. Subsymmetric ultraﬁlters Lemma The set S± k is a closed subsemigroup of γFIN± k . Lemma T(U) ∈ S± k−1 for all U ∈ S± k . Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 23 / 37

45. Do subsymmetric ultrafilters exist? Answer: It is claimed by Todorcevic, but he does not prove it in detail. Lemma The set S± k = ∅ for all k ≥ 1. Todorcevic’s hint. Let V be a cofinite ultrafilter on FIN± k . Then U = Tk−1 (V) − Tk−1 (V) + Tk−2 (V) − Tk−2 (V) + . . . + + T(V) − T(V) + V − V + T(V) − T(V) + . . . + + Tk−2 (V) − Tk−2 (V) + Tk−1 (V) − Tk−1 (V) is a subsymmetric ultrafilter. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 24 / 37

48. Gowers’ Ramsey Theorem Theorem (Gowers) For every finite partition of FIN± k there is a piece P of the partition such that (P)1 contains a partial subsemigroup of FIN± k generated by an infinite basic block sequence. We cannot ask the partial semigroup to be contained entirely in P. Consider the colourings: 1 RED if the first nonzero coordinate is positive, BLUE otherwise. Then f and −f have always different colours. 2 RED if the first and last nonzero coordinates have the same sign, BLUE otherwise. Then f + g and f − g have always different colours. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 25 / 37

52. Gowers’ Ramsey Theorem To prove Gowers’ Theorem we need the following lemma: Lemma There exists a sequence of ultraﬁlters Uj ∈ S± j (1 ≤ j ≤ k) such that for all 1 ≤ i ≤ j ≤ k: (i) Ui + Ui = Ui ; (ii) Ui + Uj = Uj + Ui = Uj ; (iii) Tj−i (Uj ) = Ui . Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 26 / 37

53. Gowers’ Ramsey Theorem A sketch of the proof: • Consider the sequence of ultraﬁlter Uj (1 ≤ j ≤ k) given by the lemma. • Let P be the piece of the partition such that P ∈ Uk. • Since Uk is subsymmetric, (P)1 ∩ −(P)1 ∈ Uk. • We can construct a block sequence {xn}n and a decreasing sequence of sets {Al n}n (1 ≤ l ≤ k) such that 1 Ak 0 = (P)1 ∩ −(P)1, Al n = Tk−l (Ak n); 2 Al n = −Al n ∈ Ul ; 3 ±xn ∈ Ak n and ±Tk−l (xn) ∈ Al n; 4 Cl,j m = {y ∈ FIN± l | ± Tk−j (xm) ± y ∈ A max{l,j} m } ∈ Ul for 1 ≤ j, l ≤ k. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 27 / 37

58. Gowers’ Ramsey Theorem We can represent the sequence {Al n}n (1 ≤ l ≤ k) with this diagram Ak 0 ⊃ Ak 1 ⊃ . . . ⊃ Ak n−2 ⊃ Ak n ∈ Uk ↓T ↓T Ak−1 0 ⊃ Ak−1 1 ⊃ . . . ⊃ Ak−1 n−2 ⊃ Ak−1 n ∈ Uk−1 ↓T ↓T ... ... ↓T ↓T A1 0 ⊃ A1 1 ⊃ . . . ⊃ A1 n−2 ⊃ A1 n ∈ U1 Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 28 / 37

59. Gowers’ Ramsey Theorem 0Tj0 (xn0 ) + 1Tj1 (xn1 ) + . . . + l Tjl (xnl ), 1 Ak 0 = (P)1 ∩ −(P)1, Al n = Tk−l (Ak n). 2 Al n = −Al n ∈ Ul . 3 ±xn ∈ Ak n and ±Tk−l (xn) ∈ Al n. ensure that +Ti (xni ), −Ti (xni ), and their sums belong to the generated subspace. 4 Cl,j m = {y ∈ FIN± l | ± Tk−j (xm) ± y ∈ A max{l,j} m } ∈ Ul . ensures we can recursively pick xn so that the sums are well-deﬁned. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 29 / 37

63. A δ-net on c0 Let k be a positive integer and let 0 < δ < 1 be such that (1 + δ)1−k = δ. Let ∆± k be the family of all maps f : N → {0, ±(1 + δ)1−k , ±(1 + δ)2−k , . . . , ±(1 + δ)−1 , ±1}, which attain at least one of the values ±1, and such that supp(f ) = {n ∈ N | f (n) = 0} is ﬁnite. ∆± k is a δ-net on Sc0 , i.e. Sc0 = f ∈∆± k Sc0 (f , δ). Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 30 / 37

66. A δ-net on c0 Consider the function φ : R → R, φ(x) = log |x| log((1 + δ)−1) . Notice that φ(±(1 + δ)h−k) = k − h. We deﬁne the map Φ : Sc0 → FIN± k by Φ(f )(n) = sign(f (n)) · max{0, k − φ(f (n)) }. Thus, if f ∈ ∆± k and f (n) = ±(1 + δ)h−k we have that Φ(f )(n) = ± max{0, h}. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 31 / 37

70. A δ-net on c0 Hence Φ|∆± k is a bijection onto FIN± k . Lemma For every f ∈ ∆± k and 0 ≤ λ ≤ 1. Φ(λ · f ) = Tj (Φ(f )) for j = min{k, φ(λ) }. Corollary For any ﬁnite partition of ∆± k , there is an inﬁnite dimensional block subspace X of c0 and there is some piece P of the partition such that SX ⊆ (P)δ. Indeed, since (1 + δ)1−k = δ, we have that Φ((A)δ) = (Φ(A))1. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 32 / 37

74. The oscillation-stability problem Theorem Let > 0 and let F be a real-valued Lipschitz function on the unit sphere of c0. Then there is an inﬁnite-dimensional subspace of c0 on the unit sphere of which F varies by at most . A sketch of the proof: • Let K be the Lipschitz constant of F. • Find k > 0 such that if (1 + δ)1−k = δ then δ · K ≤ /2. • Since the range of F is bounded, we can ﬁnitely partition into sets of diameter ≤ /2. • We conclude by applying the previous corollary. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 33 / 37

78. The importance of Gowers’ Theorem Theorem (Hindman, on finite sums) For every finite colouring of N, there is an infinite sequence {bn} ⊆ N such that the family of all finite sums of elements of {bn} is monochromatic. Theorem (Hindman, on finite unions) Let Pf be the family of all finite nonempty subsets of N. For every finite colouring of Pf , there exists an infinite sequence {bn} of disjoint subsets of N such that the family of all finite unions of finite nonempty subfamilies of {bn} is monochromatic. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 34 / 37

79. The importance of Gowers’ Theorem Theorem (Hindman, on finite sums) For every finite colouring of N, there is an infinite sequence {bn} ⊆ N such that the family of all finite sums of elements of {bn} is monochromatic. Theorem (Hindman, on finite unions) Let Pf be the family of all finite nonempty subsets of N. For every finite colouring of Pf , there exists an infinite sequence {bn} of disjoint subsets of N such that the family of all finite unions of finite nonempty subfamilies of {bn} is monochromatic. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 34 / 37

80. The importance of Gowers’ Theorem If we consider FINk instead of FIN± k , one has a Ramsey result: Theorem (Gowers) For every finite colouring of FINk, there is an infinite block sequence B of elements of FINk such that the partial semigroup generated by B is monochromatic. If we take k = 1, one has Theorem For every finite colouring of FIN1, there is an infinite block sequence B of elements of FIN1 such that the partial semigroup generated by B is monochromatic. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 35 / 37

81. The importance of Gowers’ Theorem If we consider FINk instead of FIN± k , one has a Ramsey result: Theorem (Gowers) For every finite colouring of FINk, there is an infinite block sequence B of elements of FINk such that the partial semigroup generated by B is monochromatic. If we take k = 1, one has Theorem For every finite colouring of FIN1, there is an infinite block sequence B of elements of FIN1 such that the partial semigroup generated by B is monochromatic. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 35 / 37

82. The importance of Gowers’ Theorem Theorem For every finite colouring of FIN1, there is an infinite block sequence B of elements of FIN1 such that the partial semigroup generated by B is monochromatic. Theorem (Hindman, on finite unions) Let Pf be the family of all finite nonempty subsets of N. For every finite colouring of Pf , there exists an infinite sequence {bn} of disjoint subsets of N such that the family of all finite unions of finite nonempty subfamilies of {bn} is monochromatic. Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 36 / 37

83. - Thanks for your attention - Pietro Porqueddu (University of Pisa) Gowers’ Ramsey Theorem 03/02/2017 37 / 37

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