Slow P -point Ultrafilters

Transcription

1 Slow P -point Ultrafilters Renling Jin College of Charleston jinr@cofc.edu Abstract We answer a question of Blass, Di Nasso, and Forti [2, 7] by proving, assuming Continuum Hypothesis or Martin s Axiom, that (1) there exists a P -point which is not interval-to-one and (2) there exists an interval-to-one P -point which is neither quasi-selective nor weakly Ramsey. 1 Introduction Quasi-selective ultrafilter is first introduced in [4] where it is actually called smooth ultrafilter. Quasi-selective ultrafilter is interesting because the existence of such ultrafilters is equivalent to the existence of coherent fine densities [4, Theorem 3.2] and to the existence of asymptotic numerosities [2, Corollary 5.3]. Since a selective ultrafilter is quasi-selective and a quasiselective ultrafilter is a P -point, it is natural to study whether these three classes of ultrafilters are really distinct. It is shown in [2] that, under CH, these three classes of ultrafilters are distinct. In fact, the quasi-selective nonselective ultrafilter F constructed in [2] is also weakly Ramsey. In [7] it is pointed out that both quasi-selective ultrafilters and weakly Ramsey ultrafilters are a special kind of P -points, called interval-to-one P -points. The question whether the class of P -points, the class of interval-to-one P -points, and the class of quasi-selective or weakly Ramsey ultrafilters are distinct is asked in both [2, page 1484] and [7, page 11]. Since by a celebrated result Mathematics Subject Classification 2010 Primary 03E05, 03E65 Keywords: slow ultrafilter, P -point, interval-to-one P -point, quasi-selective ultrafilter, weakly Ramsey ultrafilter Acknowledgment: This work was partially supported by a collaboration grant (ID: ) from Simons Foundation. 1

2 of S. Shelah [10] it is consistent that there exist no P -points, we can only discuss the differences between these ultrafilters under some set theoretical assumptions beyond ZFC such as, for example, CH (Continuum Hypothesis), MA (Martin s Axiom), etc. In this paper we show that these classes of ultrafilters are distinct assuming either CH or MA. Rapid ultrafilters are another class of ultrafilters closely related to selective ultrafilters and P -points. It is shown in [2, Corollary 1.6] that an ultrafilter F is selective if and only if it is rapid and quasi-selective. In [12] it is shown that an ultrafilter F is rapid if and only if the intersection of F with any tall summable ideal is non-empty. Therefore, a non-selective quasi-selective ultrafilter F must have empty intersection with some tall summable ideal I f determined by a non-increasing function f : N [0, 1]. As an antonym for the word rapid we call an ultrafilter f-slow if it is disjoint from I f. In this paper we want to find P -point ultrafilters which are not interval-toone and interval-to-one P -point ultrafilters which are neither quasi-selective nor weakly Ramsey. We look for these ultrafilters among all f-slow ultrafilters for some f. Indeed, we do find them by carefully controlling the speeds of their slow-ness. In Section 2 we state the definitions of various involved ideals and filters. Although many of these definitions are well-known to a set theorist, we include the definitions here to make the paper somewhat self-contained. In Section 3 we prove our results under CH and in Section 4 we do the same under MA 1. Some open problems are in Section 5. 2 Summable ideals and slow ultrafilters By an interval [a, b] we mean an interval of integers {x Z a x b}. Occasionally, we use the interval notation [0, 1] for the unit interval of reals. Hope that won t cause confusion. Let N be the set of all positive integers. Definition 2.1 A family of sets I P(N) is a free ideal if 1. N I, 2. F I for every finite or empty subset F of N, 3. A I and B A imply B I, and 1 In fact, we need only MA for σ-centered partial orders. 2

3 4. A B I for any A, B I. Definition 2.2 A family of sets F P(N) is a free filter if 1. F, 2. N \ F F for every finite or empty subset F of N, 3. A F and B A imply B F, and 4. A B F for any A, B F. A free filter F is called an ultrafilter if A F or N \ A F for any A N. Since all ideals and filters considered in this paper are free, we will omit the word free from now on. Clearly, I is an ideal if and only if F = {N \ A A I} is a filter. In that case, I and F are dual of each other. Definition 2.3 Let f : N [0, 1] be a non-increasing function such that f(n) =. The following set I f is called the summable ideal determined n=1 by f where I f := { A N } f(n) <. n A The summable ideal I f is tall if lim n f(n) = 0. Notice that all non-tall summable ideals are the same which is the ideal of all finite subsets of N. For notational convenience we will frequently use the following expression. S(f, A) := n A f(n). (1) Definition 2.4 Let I f be a summable ideal determined by f. An ultrafilter F is called f-slow if F contains the dual filter of I f. Notice that an ultrafilter F is f-slow if and only if S(f, A) = for all A F. 3

4 Definition 2.5 Let A N be an infinite set and g : A N. The function g is called finite-to-one if g 1 (n) is a finite set for every n N. An ultrafilter F is called a P -point if for every function g : N N there exists an A F such that g A is either a constant function or a finite-to-one function. Notice that a function g : N N can be viewed as a partition {g 1 (n) n N} of N and a partition P = {B n n N} of N can be viewed as a function g P with g P (x) = n if and only if x B n. Definition 2.6 Let A N be an infinite set and g : A N. The function g is called interval-to-one if the intervals in C = {[min(g 1 (n)), sup(g 1 (n))] n N and g 1 (n) } are pairwise disjoint. A P -point F is said to be interval-to-one if for every function g : N N, there is an A F such that g A is interval-to-one. Notice that the collection C above can contain an infinite interval when {n g 1 (n) } is a finite set. If sup S =, the interval [a, sup S] means [a, ). Otherwise, sup(g 1 (n)) = max(g 1 (n)). Definition 2.7 An ultrafilter F is selective if for every function g : N N there is an A F such that g A is either a constant function or a one-to-one function. Definition 2.8 An ultrafilter F is said to be quasi-selective if for every function g : N N with g(n) n for all n N, there exists an A F such that g A is a non-decreasing function. Let X be a set and [X] 2 := {{a, b} a, b X and a b}. Definition 2.9 An ultrafilter F is Ramsey if for every function c : [N] 2 {0, 1} there is an A F such that range(c [A] 2 ) = 1. The set A above is called a c-homogeneous set. Definition 2.10 An ultrafilter F is said to be weakly Ramsey if for every function c : [N] 2 {0, 1, 2} there is an A F such that range(c [A] 2 ) 2. 4

5 It is a well-known fact that an ultrafilter is selective if and only if it is Ramsey. A selective ultrafilter is quasi-selective as well as weakly Ramsey, and a quasi-selective ultrafilter or a weakly Ramsey ultrafilter is an interval-to-one P -point [2, 7]. Definition 2.11 An ultrafilter F is rapid if for every finite-to-one function g : N N, there exists an A F such that g 1 (n) A n for all n N. It is shown in [12] that if F is a rapid ultrafilter and I f is a tall summable ideal determined by f, then F I f. We say that A is almost a subset of B, denoted by A B, if A \ B is a finite set. The reader may find other basic information on ultrafilters in [1, 3] and basic information on set theory, especially Continuum Hypothesis and Martin s Axiom in [8, 9]. 3 Assuming CH deter- Theorem 3.1 Assume that CH holds. For each summable ideal I f mined by f, there exists an f-slow P -point. Proof: By CH we can list all functions from N to N as {g α α < ℵ 1 }. We define sets A α inductively on all ordinals α < ℵ 1 such that for any α < < ℵ 1 1. A α I f, 2. A A α, and 3. either A α+1 gα 1 (n) for some n N or A α+1 gα 1 (n) < for every n N. The theorem follows from the construction because the ultrafilter F generated by {A α α < ℵ 1 } is an f-slow P -point. Indeed, {A α α < ℵ 1 } has FIP (finite intersection property) by 2. Also F is a P -point by 3. For f-slowness, if C I f, then {C, N \ C} is a partition of N, and hence N \ C A α for some α < ℵ 1 by 1. and 3. We start with A 0 = N. Suppose that {A α α < δ} have been constructed for some δ < ℵ 1 to satisfy 1., 2., and 3. when ℵ 1 is replaced by δ. Case 1: δ is a limit ordinal. 5

6 Let {α n < δ n N} be an increasing sequence of ordinals cofinal in δ. m For each m N, we have A αn I f by 1. and 2. For each m N let n=1 B m = {a i n m i < n m+1 } be a finite increasing sequence of elements in m such that S(f, B m ) 1 (S(f, B m ) is defined in (1)). The sequence n=1 A αn B m can be chosen such that a nm 1, the last element in B m 1, is less than a nm, the first element in B m, for every m N. Let A δ := m=1 B m. The existence of B m is guaranteed due to the fact that S(f, m n=1 A α n ) =. Clearly, {A α α < δ + 1} satisfies 1. and 2. when ℵ 1 is replaced by δ + 1. The condition 3. is vacuous. Case 2: δ = + 1. Let h = g A. If there is an n N such that h 1 (n) I f, let A δ = h 1 (n). Clearly, {A α α < δ + 1} satisfies 1., 2., and 3. when ℵ 1 is replaced by δ + 1. Hence we can assume that h 1 (n) I f for every n N. Let n 0 = 1 and A 0 = A. We now construct an increasing sequence {n i } and a decreasing sequence {A i } of sets not in I f by induction such that S(f, D i ) 1 where D i = A i 1 A i = Ai 1 [n i 1, n i 1] and \ {h 1 (n) h 1 (n) D i }. Notice that there are only finitely many n with h 1 (n) D i. Hence A i I f if A i 1 I f. Suppose we have had n k 1 and A k 1 I f. Let n k be large enough such that S(f, A k 1 [n k 1, n k 1]) 1. The number n k exists because A k 1 I f. Let A k = Ak 1 \ {h 1 (n) h 1 (n) D k } where D k = A k 1 [n k 1, n k 1]. This completes the construction. Let A δ := D i. Then A δ I f because S(f, A δ ) = S(f, D i ) = i N. Clearly, A δ A. For each n N, let k be the smallest such that D k g 1 (n) or k = 1 if D k g 1 (n) = for every k N. Hence D i g 1 (n) = D i h 1 (n) = for every i > k. This shows that A δ g 1 (n) i=1 6

7 [1, n k 1]. Therefore, we have that {A α α < δ + 1} satisfies Conditions 1., 2., and 3. when ℵ 1 is replaced by δ + 1. Notice that if lim n f(n) > 0, then f-slow P -point is just the ordinary P -point. Hence Theorem 3.1 is only interesting when I f is tall. Theorem 3.2 Let f(n) = 1/n. There does not exist interval-to-one f-slow P -point. Proof: Assume by contrary that F is an interval-to-one f-slow P -point. Let a 0 = 0, a n+1 = a n + n 2 for any n 0. Notice that [a n, a n+1 1] is an interval of length n 2. Let B m := {a n + m + in i = 0, 1,..., n 1}. n=m Notice that the set {a n + m + in i = 0, 1,..., n 1} is the m-th arithmetic progression of length n and difference n in [a n, a n+1 1]. Therefore, P = {B m m N} is a partition of N. Since a n+1 = n 2 + (n 1) = 1 n(n + 1)(2n + 1), 6 we have that S(f, B m ) = n=m S(f, B m [a n, a n+1 1]) n=m n a n 1 + n=m+1 3 (n 1) 2 <. Hence B m I f for every m N. Since F is an interval-to-one f-slow P - point, there exists a set A F such that intervals [min(a B), max(a B)] for all B P are pairwise disjoint. Claim A [a n, a n+1 1] 2n. Proof of Claim: Notice that B m [a n, a n+1 1] = for all m > n. Suppose there are k sets B m1, B m2,..., B mk P for some k n such that B mi A [a n, a n+1 1] for i = 1, 2,..., k. Let l i = min(b mi A [a n, a n+1 1]) and u i = max(b mi A [a n, a n+1 1]). Notice that [l i, u i ] A 7

8 contains at most e i := 1 + (u i l i )/n elements from B mi and contains no elements from any other B mj for j i. So n 2 k (u i l i + 1) = i=1 k k (n(e i 1) + 1) = n e i kn + k, i=1 i=1 which implies that A [a n, a n+1 1] k e i 1 n (n2 + kn k) 1 n (2n2 k) 2n. i=1 This completes the proof of the claim. Now the theorem follows from the claim because 2n S(f, A) = S(f, A [a n, a n+1 1]) <. a n n=1 n=1 This contradicts the assumption that A I f. Notice that Theorem 3.2 is a consequence of ZFC. Corollary 3.3 Assume that CH holds. There exists a non-interval-to-one P -point. Proof: Let f(n) = 1/n. By Theorem 3.1 there is an f-slow P -point F. By Theorem 3.2 the P -point F cannot be interval-to-one. In the proof of the next theorem we need to use three existing results. The first one is a result in [2, Proposition 1.7] which guarantees that if F is a quasi-selective ultrafilteer and h(n) = 2 n, then for every function h : N N with h (n) h(n) for all n N, there exists a set A F such that h A is non-decreasing. The second result we need is so called Erdős-Szekeres Theorem [6], which says that every finite sequence of real numbers of length n contains a monotonic subsequence of length at least n + 1. In the proof of next theorem we need only a consequence for notational convenience that every sequence of length n 2 contains a monotonic subsequence of length at least n. The third result we need is a lower bound for Ramsey numbers by P. Erdő [5] which is later improved in [11]. Recall that [X] 2 := {{a, b} a, b 8

9 X and a b}. For any r N let R(r) be the smallest positive integer n such that for any c : [X] 2 {0, 1} where X = n there is a set X X such that X r and c [X] 2 is a constant function. It is shown in [11] that for all sufficiently large r. R(r) > Cr2 r 2 where C = 2 e (2) Theorem 3.4 Assume that CH holds. There exists an interval-to-one P - point, which is neither quasi-selective nor weakly Ramsey. Proof: Let a 0 = 1 and a n = 2 n! for n 1. Let p : N N be such that p(x) = n if and only if a n x < a n+1. Let f : N [0, 1] be such that f(x) = 1/2 (n 1)! if x p 1 (n). Notice a n+1 a n that f is non-increasing, lim f(n) = 0, and S(f, N) =. n 2 (n 1)! n=1 Let I f be the tall summmable ideal determined by f. Notice also that if A p 1 (n) 2 (n 1)! for infinitely many n, then A I f. The ultrafilter F that we construct will be f-slow. By CH we can list all functions from N to N as a sequence {g α α < ℵ 1 }. We construct a sequence {A α α < ℵ 1 } of sets by induction such that for any α < < ℵ 1 1. A α I f, 2. A A α, 3. {n N A α p 1 (n) > a k n 1} is infinite for each k N, 4. either A α+1 gα 1 (m) for some m N or intervals in {[min(a α+1 gα 1 (m)), max(a α+1 gα 1 (m))] m N} are pairwise disjoint. Condition 1 is listed for convenience only. It is actually a consequence of Condition 3. We start with A 0 = N. Notice that Condition 3 is true for A 0 because if k 1 is fixed, then a n+1 a n > a k n 1 2 (n 1)! for all sufficiently large n N. Conditions 2. and 4. are vacuous for A 0. 9

10 Assume that we have obtained all sets A α for α < δ for some δ < ℵ 1 such that the four properties above are satisfied when ℵ 1 is replaced by δ. Case 1: δ is a limit ordinal. Let {α m m N} be an increasing sequence of ordinals below δ and cofinal in δ. Without loss of generality we can assume that A αm+1 A αm for every m N. Let n 1 be the smallest positive integer such that A α1 p 1 (n 1 ) > a 1 n 1 1. For each m > 1 let n m > n m 1 be the smallest integer such that A αm p 1 (n m ) > a m n m 1. The existence of n m is guaranteed by Condition 3. Now let A δ := (A αm p 1 (n m )). m=1 Clearly, Condition 2 is true because for a fixed m 0, m=m 0 (A αm p 1 (n m )) A αm0. Hence A δ A α for any α < δ. Condition 3 is true because for each fixed k, A δ p 1 (n m ) > a m n m 1 a k n m 1 for all m > k. Condition 4 is vacuous. Case 2: δ is a successor ordinal + 1. By Erdős Szekeres Theorem we have that every sequence of real numbers with n 2 terms contains a monotonic subsequence with n terms. First we assume that there exists an m N such that for each k N there are infinitely many n with A g 1 (m) p 1 (n) > a k n 1. Then we set A δ := A g 1 (m). It is easy to check that {A α α < δ + 1} satisfies Conditions 1 4 above when ℵ 1 is replaced by δ + 1. Hence we can now assume that for every m, there is a k m and n m such that A g 1 (m) p 1 (n) a km n 1 (3) for every n n m. Let S = {n A p 1 (n) > a n 1 }. We construct an increasing sequence n 1 < n 2 < in S and sets D ni p 1 (n i ) A inductively such that a. D ni > a i n i 1, 10

11 b. g D ni is monotonic, and c. g (x) g (y) for any x D ni and y D nj when i j. Let n 1 = min{n S A p 1 (n) > a 2 n 1}. By Erdős Szekeres Theorem the sequence {g (x) x A p 1 (n 1 )} contains a monotonic subsequence {g (x) x D n1 } with D n1 > a n 1. Notice that g D n1 is interval-to-one because it is monotonic. Suppose we have constructed n 1 < n 2 < < n t and D ni A p 1 (n i ) for i = 1, 2,..., t such that D ni > a i n i 1, g is monotonic on D ni, and g (x) g (y) for any x D ni and y D nj when i j for all 1 i, j t. For constructing n t+1 and D nt+1 let X := {m D ni g 1 (m) for some i t}. Notice that X a nt+1. Let k = max{k m m X} and n = max{n m m X} where k m and n m are defined in (3). Recall that A g 1 (m) p 1 (n) a km n 1 a k n 1 for all m X and all n n. Let k = max{2p + 3, k + 2}. By Condition 3 we can find an n t+1 max{n t + 1, n} + 1 in S such that A p 1 (n t+1 ) > a k n t+1 1. Let Then we have A = (A \ {g 1 (m) m X}) p 1 (n t+1 ). A > a k n t+1 1 a nt+1 a k nt+1 1 a k n t+1 1 a k+1 n t+1 1 a k 1 n t+1 1 a 2(t+1) n t+1 1. By Erdős Szekeres Theorem we can find an D nt+1 A such that D nt+1 > a t+1 n t+1 1 and g is monotonic on D nt+1. Clearly, a. and b. are satisfied. The condition c. is true because all function values of g on D nt+1 are not in X. Now set A δ := D ni. Clearly, A δ A. Hence Condition 2 is true. Since for a fixed k, i=1 A δ p 1 (n i ) = D i > a i n i 1 a k n i 1 for all i k, Condition 3 is true. Condition 4 is true because g is monotonic on each D ni, and the range of g D i and the range of g D j are pairwise disjoint for any i j. 11

12 Let F = {U N A α U for some α < ℵ 1 }. Clearly, F is a filter. For any B N, there must be a function g α such that g α (x) = 1 if x B and g α (x) = 2 if x B. By Condition 4 we have that A α+1 gα 1 (1), which implies B F, or A α+1 gα 1 (2), which implies N \ B F. Hence F is an ultrafilter. Also by Condition 4 F is an interval-to-one P -point. Since A α I f by Condition 1, the ultrafilter F is f-slow. We now show that F is neither quasi-selective nor weakly Ramsey. Suppose that F is a quasi-selective ultrafilter. Let h(n) = 2 n. By [2, Proposition 1.7] any function h : N N with h (n) h(n) for all n N is F-equivalent to a non-decreasing function. For each x [a n, a n+1 1] let h (x) = a n+1 (x a n ). Then h (x) a n+1 = 2 (n+1)! 2 2n! = h(a n ) h(x). Hence there is an A F such that h is non-decreasing on A, which implies that A [a n, a n+1 1] contains at most one element. So we have that S(f, A) n=1 1 2 (n 1)! <, which contradicts that F is f-slow. Now we show that F is not weakly Ramsey. By (2) we have that R(2 (n 2)! ) > C2 (n 2)! 2 2(n 2)!/2 for all sufficiently large n. Let X n = [a n, a n+1 1]. Since C2 (n 2)! 2 2(n 2)!/2 lim =, n 2 (n+1)! 2 n! there is a function c n : [X n ] 2 {0, 1} for all sufficiently large n such that max{ X X X n is c n -homogeneous} < 2 (n 2)!. Assume by the contrary that F is weakly Ramsey. For every {a, b} [N] 2, define 0 if p(a) = p(b) = n and c n (a) = c n (b) c({a, b}) = 1 if p(a) = p(b) = n and c n (a) c n (b) 2 if p(a) p(b). Then there exists a set A F such that c takes only two values among {0, 1, 2}. If the range of c on [A] 2 is {0, 1}, then A p 1 (n) for some n. This 12

13 is impossible because A is infinite. If the range of c on [A] 2 is {1, 2}, then A p 1 (n) 2 because c n can have only two values. Hence S(f, A) < n=1 2 2 (n 1)! <, contradicting that A I f. If the range of c on [A] 2 is {0, 2}, then A p 1 (n) is c n -homogeneous, which implies that there is an n 0 N such that A p 1 (n) < 2 (n 2)! for all n n 0. Hence S(f, A) S(f, A [1, a n0 1]) + 2(n 2)! <, 2 (n 1)! n=n 0 which again contradicts that A I f. This completes the proof of the theorem. 4 Under MA deter- Theorem 4.1 Assume that MA holds. For each summable ideal I f mined by f, there exists an f-slow P -point. Proof: The steps of the proof for this theorem is similar to those of Theorem 3.1. So we borrow notation from there. The main difference is that the list {g α α < 2 ℵ 0 } of all functions from N to N has a length of 2 ℵ 0, which may be greater than ℵ 1 and therefore, our construction of the sequence of sets {A α α < 2 ℵ 0 } must also have a length of 2 ℵ 0. More specifically, we want to construct sets A α inductively on all ordinals α < 2 ℵ 0 such that for any α < < 2 ℵ 0 1. A α I f, where I f is a fixed summable ideal determined by f, 2. A A α, and 3. either A α+1 gα 1 (n) for some n N or A α+1 gα 1 (n) < for every n N. The steps of the construction of A δ for successor ordinals δ = + 1 are identical to the correspondent part in the proof of Theorem 3.1. In the case of δ being a limit ordinal we find A δ by MA. Notice that in this case Condition 3 is vacuous. 13

14 Suppose that δ < 2 ℵ 0 is a limit ordinal and A δ = {A α α < δ} has been constructed so that it satisfies Condition 1, 2, and 3 when 2 ℵ 0 is replaced by δ. In order to find A δ we define a partial order P such that P = {(n, s, F ) n N, s [1, n], F A δ, and F is finite}. For any p = (n p, s p, F p ), q = (n q, s q, F q ) P, define p q (or p is stronger than q) if 1. n p n q, 2. s p [1, n q ] = s q, 3. F q F p, 4. s p [n q + 1, n p ] A for any A F q, 5. S(f, s p [n q + 1, n p ]) 1. If n p = n q and s p = s q, then (n p, s p, F p F q ) is a common lower bound of p and q. Hence (P, ) is σ-centered. So (P, ) has c.c.c. Let D A,n = {p P n p n and A F p } for each n N and A A δ. It is routine to check that D A,n is dense in P. Since D = {D A,n A A δ and n N} has less than 2 ℵ 0 members, there exists, by MA, a D generic filter G P, i.e., a filter G G such that G D for all D D. Let A δ := {s p p G}. Then A δ A α for any α < δ because G D Aα,n, and A δ I f because there are increasing sequence p 1 > p 2 >... in G with n p1 < n p2 < and thus S(f, A δ [n pi + 1, n pj+1 ]) 1 for i = 1, 2,.... Corollary 4.2 Assume MA. There exists a non-interval-to-one P -point. Proof: Notice that Theorem 3.2 is a result of ZFC. Therefore, the existence of a 1/n-slow non-interval-to-one P -point follows from Theorem 4.1 and Theorem 3.2. Theorem 4.3 Assume that MA holds. There exists an interval-to-one P - point, which is neither quasi-selective nor weakly Ramsey. 14

15 Proof: Similar to the proof of Theorem 4.1 the proof of this theorem is parallel to the proof of Theorem 3.4. So we borrow also notation from there. The only step different from that of the proof of Theorem 3.4 is the construction of the set A δ when δ is a limit ordinal. Suppose that δ < 2 ℵ 0 is a limit ordinal and A δ = {A α α < δ} has been constructed. Notice that the letter p is already used for a function defined in the first line of the proof of Theorem 3.4. For obtaining the set A δ let Q = {(s, D, F, k) s N is finite, D n s p 1 (n), F A δ is finite, and k N}. For any q = (s q, D q, F q, k q ), q = (s q, D q, F q, k q ) Q define that q q if 1. s q [1, max s q ] = s q, 2. D q p 1 (n) = D q p 1 (n) for each n s q, 3. D q p 1 (n) A for each n s q \ s q and A F q, 4. k q k q, 5. D q p 1 (n) > a k q n 1 for each n s q \ s q. It is routine to check that (Q, ) is σ-centered. Let D = {D A,n,k A A δ and n, k N} where D A,n,k = {q Q max s q > n, A F q, and k q > k}. It is easy to see that D A,n,k is dense in Q and D has less than 2 ℵ 0 members. By MA there is a D generic filter G Q. Let A δ := {D q q G}. This completes the construction of A δ. It is now routine to check that A δ is the set we want. 5 Open questions An interval-to-one P -point can never be 1/n-slow by Theorem 3.2. The interval-to-one P -point which is neither quasi-selective nor weakly Ramsey in 15

16 Theorem 3.4 is f-slow for a function f with a very slow speed of approaching zero. It may be interesting to characterize the slow-ness of f which allows an f-slow interval-to-one P -point to exist. In particular we don t know the answer to the follwing question. Question 5.1 Assume that CH holds. P -point exist? Can 1/ log(n)-slow interval-to-one It is interesting to explore the relative consistency strength among these ultrafilters. Question 5.2 Is it consistent, relative to ZFC, that there exist P -points but not interval-to-one P -points? Is it consistent, relative to ZFC, that there exist interval-to-one P -points but not quasi-selective ultrafilters? Is it consistent, relative to ZFC, that there exist interval-to-one P -points but not weakly Ramsey ultrafilters? References [1] A. Blass, Combinatorial cardinal characteristics of the continuum, in: M. Foreman, A. Kanamori (Eds.), Handbook of Set Theory, Springer- Verlag, 2010, [2] A. Blass, M. Di Nasso, and M. Forti, Quasi-selective ultrafilters and asymptotic numerosities, Advances in Mathematics, vol. 231 (2012), [3] D. Booth, Ultrafilters on a countable set, Ann. Math. Log. 2 ( ), [4] M. Di Nasso, Fine asymptotic densities for sets of natural numbers, Proc. Amer. Math. Soc. 138 (2010) [5] P. Erdős, Some remarks on the theory of graphs, Bull. Amer. Math. Sot. 53 (1947), [6] P. Erd s and G. Szekeres, A combinatorial problem in geometry, Compositio Mathematica, 2: , (1935) 16

17 [7] M. Forti, Quasi-selective and weakly Ramsey ultrafilters, arxiv: [8] K. Kunen, Set Theory an introduction to independence proofs, North Holland, Amsterdam, [9] K. Kunen, Set Theory, Mathematical logic and foundations, vol. 34, College Publications, [10] S. Shelah, Proper and Improper Forcing, second ed., Springer-Verlag, [11] J. Spencer, (1975), Ramsey s theorem a new lower bound, Journal of Combinatorial Theory Ser. A, 18, [12] P. Vojtáš, On ω and absolutely divergent series, Topology Proceedings 19, ,