TY - JOUR
T1 - The Ramsey property implies no mad families
AU - Schrittesser, David
AU - Törnquist, Asger
PY - 2019
Y1 - 2019
N2 - We show that if all collections of infinite subsets of N have the Ramsey property, then there are no infinite maximal almost disjoint (mad) families. The implication is proved in Zermelo–Fraenkel set theory with only weak choice principles. This gives a positive solution to a long-standing problem that goes back to Mathias [A. R. D. Mathias, Ann. Math. Logic 12, 59–111 (1977)]. The proof exploits an idea which has its natural roots in ergodic theory, topological dynamics, and invariant descriptive set theory: We use that a certain function associated to a purported mad family is invariant under the equivalence relation E0 and thus is constant on a “large” set. Furthermore, we announce a number of additional results about mad families relative to more complicated Borel ideals.
AB - We show that if all collections of infinite subsets of N have the Ramsey property, then there are no infinite maximal almost disjoint (mad) families. The implication is proved in Zermelo–Fraenkel set theory with only weak choice principles. This gives a positive solution to a long-standing problem that goes back to Mathias [A. R. D. Mathias, Ann. Math. Logic 12, 59–111 (1977)]. The proof exploits an idea which has its natural roots in ergodic theory, topological dynamics, and invariant descriptive set theory: We use that a certain function associated to a purported mad family is invariant under the equivalence relation E0 and thus is constant on a “large” set. Furthermore, we announce a number of additional results about mad families relative to more complicated Borel ideals.
KW - Borel ideals
KW - Invariant descriptive set theory
KW - Maximal almost disjoint families
KW - Ramsey property
UR - http://www.scopus.com/inward/record.url?scp=85072315429&partnerID=8YFLogxK
U2 - 10.1073/pnas.1906183116
DO - 10.1073/pnas.1906183116
M3 - Journal article
C2 - 31467168
AN - SCOPUS:85072315429
SN - 0027-8424
VL - 116
SP - 18883
EP - 18887
JO - Proceedings of the National Academy of Sciences of the United States of America
JF - Proceedings of the National Academy of Sciences of the United States of America
IS - 38
ER -