Mixed sequences Antonio Avil´esand Stevo Todorcevic Universidad de Murcia Luminy 2012 1 incompatible, no subsequence belongs to two classes at the same time 2 hereditary, if a subsequence belongs to Ci , any further subsequence belongs as well. 3 analytic, when the set of subsequences is viewed inside 2N. The main question In how many ways can these classes of subsequences can be mixed in the sequence (xk )? C1;:::;Cn classes of subsequences which are For expositional ease, we will consider a very particular case. Mixed sequences We are given a sequence (xk ) and 1 incompatible, no subsequence belongs to two classes at the same time 2 hereditary, if a subsequence belongs to Ci , any further subsequence belongs as well. 3 analytic, when the set of subsequences is viewed inside 2N. The main question In how many ways can these classes of subsequences can be mixed in the sequence (xk )? For expositional ease, we will consider a very particular case. Mixed sequences We are given a sequence (xk ) and C1;:::;Cn classes of subsequences which are 2 hereditary, if a subsequence belongs to Ci , any further subsequence belongs as well. 3 analytic, when the set of subsequences is viewed inside 2N. The main question In how many ways can these classes of subsequences can be mixed in the sequence (xk )? For expositional ease, we will consider a very particular case. Mixed sequences We are given a sequence (xk ) and C1;:::;Cn classes of subsequences which are 1 incompatible, no subsequence belongs to two classes at the same time 3 analytic, when the set of subsequences is viewed inside 2N. The main question In how many ways can these classes of subsequences can be mixed in the sequence (xk )? For expositional ease, we will consider a very particular case. Mixed sequences We are given a sequence (xk ) and C1;:::;Cn classes of subsequences which are 1 incompatible, no subsequence belongs to two classes at the same time 2 hereditary, if a subsequence belongs to Ci , any further subsequence belongs as well. The main question In how many ways can these classes of subsequences can be mixed in the sequence (xk )? For expositional ease, we will consider a very particular case. Mixed sequences We are given a sequence (xk ) and C1;:::;Cn classes of subsequences which are 1 incompatible, no subsequence belongs to two classes at the same time 2 hereditary, if a subsequence belongs to Ci , any further subsequence belongs as well. 3 analytic, when the set of subsequences is viewed inside 2N. For expositional ease, we will consider a very particular case. Mixed sequences We are given a sequence (xk ) and C1;:::;Cn classes of subsequences which are 1 incompatible, no subsequence belongs to two classes at the same time 2 hereditary, if a subsequence belongs to Ci , any further subsequence belongs as well. 3 analytic, when the set of subsequences is viewed inside 2N. The main question In how many ways can these classes of subsequences can be mixed in the sequence (xk )? Mixed sequences We are given a sequence (xk ) and C1;:::;Cn classes of subsequences which are 1 incompatible, no subsequence belongs to two classes at the same time 2 hereditary, if a subsequence belongs to Ci , any further subsequence belongs as well. 3 analytic, when the set of subsequences is viewed inside 2N. The main question In how many ways can these classes of subsequences can be mixed in the sequence (xk )? For expositional ease, we will consider a very particular case. Definition A sequence (xk ) is ~p-saturated if every subsequence contains a pi -sequence for some i. Definition A sequence (xk ) is a ~p-sequence if it is ~p-saturated and it cannot be writen as a finite union of ~q-saturated sequences for j~qj < n. Now consider ~p = (p1;:::;pn), j~pj = n. The basis of `1 ⊕ `2 is not a (1;2)-sequence. The basis of `1(`2) is a (1;2)-sequence. Remark: Any ~p-saturated sequence is the union of ~q-saturated sequences for ~q ⊂~p. Mixed sequences of `p-sequences A p-sequence is a sequence (xk ) equivalent to the canonical basis of `p. it is ~p-saturated and it cannot be writen as a finite union of ~q-saturated sequences for j~qj < n. Definition A sequence (xk ) is ~p-saturated if every subsequence contains a pi -sequence for some i. Definition A sequence (xk ) is a ~p-sequence if The basis of `1 ⊕ `2 is not a (1;2)-sequence. The basis of `1(`2) is a (1;2)-sequence. Remark: Any ~p-saturated sequence is the union of ~q-saturated sequences for ~q ⊂~p. Mixed sequences of `p-sequences A p-sequence is a sequence (xk ) equivalent to the canonical basis of `p. Now consider ~p = (p1;:::;pn), j~pj = n. it is ~p-saturated and it cannot be writen as a finite union of ~q-saturated sequences for j~qj < n. Definition A sequence (xk ) is a ~p-sequence if The basis of `1 ⊕ `2 is not a (1;2)-sequence. The basis of `1(`2) is a (1;2)-sequence. Remark: Any ~p-saturated sequence is the union of ~q-saturated sequences for ~q ⊂~p. Mixed sequences of `p-sequences A p-sequence is a sequence (xk ) equivalent to the canonical basis of `p. Now consider ~p = (p1;:::;pn), j~pj = n. Definition A sequence (xk ) is ~p-saturated if every subsequence contains a pi -sequence for some i. it is ~p-saturated and it cannot be writen as a finite union of ~q-saturated sequences for j~qj < n. The basis of `1 ⊕ `2 is not a (1;2)-sequence. The basis of `1(`2) is a (1;2)-sequence. Remark: Any ~p-saturated sequence is the union of ~q-saturated sequences for ~q ⊂~p. Mixed sequences of `p-sequences A p-sequence is a sequence (xk ) equivalent to the canonical basis of `p. Now consider ~p = (p1;:::;pn), j~pj = n. Definition A sequence (xk ) is ~p-saturated if every subsequence contains a pi -sequence for some i. Definition A sequence (xk ) is a ~p-sequence if and it cannot be writen as a finite union of ~q-saturated sequences for j~qj < n. The basis of `1 ⊕ `2 is not a (1;2)-sequence. The basis of `1(`2) is a (1;2)-sequence. Remark: Any ~p-saturated sequence is the union of ~q-saturated sequences for ~q ⊂~p. Mixed sequences of `p-sequences A p-sequence is a sequence (xk ) equivalent to the canonical basis of `p. Now consider ~p = (p1;:::;pn), j~pj = n. Definition A sequence (xk ) is ~p-saturated if every subsequence contains a pi -sequence for some i. Definition A sequence (xk ) is a ~p-sequence if it is ~p-saturated The basis of `1 ⊕ `2 is not a (1;2)-sequence. The basis of `1(`2) is a (1;2)-sequence. Remark: Any ~p-saturated sequence is the union of ~q-saturated sequences for ~q ⊂~p. Mixed sequences of `p-sequences A p-sequence is a sequence (xk ) equivalent to the canonical basis of `p. Now consider ~p = (p1;:::;pn), j~pj = n. Definition A sequence (xk ) is ~p-saturated if every subsequence contains a pi -sequence for some i. Definition A sequence (xk ) is a ~p-sequence if it is ~p-saturated and it cannot be writen as a finite union of ~q-saturated sequences for j~qj < n. The basis of `1(`2) is a (1;2)-sequence. Remark: Any ~p-saturated sequence is the union of ~q-saturated sequences for ~q ⊂~p. Mixed sequences of `p-sequences A p-sequence is a sequence (xk ) equivalent to the canonical basis of `p. Now consider ~p = (p1;:::;pn), j~pj = n. Definition A sequence (xk ) is ~p-saturated if every subsequence contains a pi -sequence for some i. Definition A sequence (xk ) is a ~p-sequence if it is ~p-saturated and it cannot be writen as a finite union of ~q-saturated sequences for j~qj < n. The basis of `1 ⊕ `2 is not a (1;2)-sequence. Remark: Any ~p-saturated sequence is the union of ~q-saturated sequences for ~q ⊂~p. Mixed sequences of `p-sequences A p-sequence is a sequence (xk ) equivalent to the canonical basis of `p. Now consider ~p = (p1;:::;pn), j~pj = n. Definition A sequence (xk ) is ~p-saturated if every subsequence contains a pi -sequence for some i. Definition A sequence (xk ) is a ~p-sequence if it is ~p-saturated and it cannot be writen as a finite union of ~q-saturated sequences for j~qj < n. The basis of `1 ⊕ `2 is not a (1;2)-sequence. The basis of `1(`2) is a (1;2)-sequence. Mixed sequences of `p-sequences A p-sequence is a sequence (xk ) equivalent to the canonical basis of `p. Now consider ~p = (p1;:::;pn), j~pj = n. Definition A sequence (xk ) is ~p-saturated if every subsequence contains a pi -sequence for some i. Definition A sequence (xk ) is a ~p-sequence if it is ~p-saturated and it cannot be writen as a finite union of ~q-saturated sequences for j~qj < n. The basis of `1 ⊕ `2 is not a (1;2)-sequence. The basis of `1(`2) is a (1;2)-sequence. Remark: Any ~p-saturated sequence is the union of ~q-saturated sequences for ~q ⊂~p. if there is a bijection e : N −! N such that (xk )k2A contains a pi -subsequence , (ye(k))k2A contains a pi -sequence. Definition 3 Two ~p-sequences (xk ) and (yk ) are equimixed Equimixed sequences ~p = (p1;:::;pn), j~pj = n.