Independence and almost disjointness

Dr Vera Fisher, Kurt Gödel Research Center for Mathematical Logic, University of Vienna. Part of the Logic Seminar Series.

The cardinal characteristics of the real line arise from various combinatorial properties of the reals.  Their study has already a long history and is closely related to the development of major forcing techniques, among which are template iterations (see [9]), matrix iterations (see [2,4]), and creature forcing (see [8]).  An excellent introduction to the subject can be found in [1].

Two of classical combinatorial cardinal characteristics of the continuum are the almost disjointness and independence numbers.  An infinite family A of infinite subsets of N, whose elements have pairwise finite intersections and which is maximal under inclusion, is called a maximal almost disjoint family.  The minimal size of such a family is denoted mathfrak{a} and is referred to as the almost disjointness number.  A family A of infinite subsets of N with the property that for any two finite disjoint subfamilies F and G, the set (Union F) \ (Union G) is infinite is said to be independent.  The minimal size of a maximal independent family is denoted mathfrak{i} and is referred to as the independence number.  The characteristics mathfrak{a} and mathfrak{i} are among those for which there are no other known upper bounds apart from the cardinality of the continuum.  It is well known that consistently mathfrak{a} < mathfrak{i}, however both the consistency of mathfrak{i} < mathfrak{a} and the inequality mathfrak{a} <= mathfrak{i} remain open.

In this talk, we will outline some of the major properties of independence and almost disjointness, describe recent results (see for example [3, 5, 6, 7]) and point out further interesting open problems.

[1] A. Blass, Combinatorial cardinal characteristics of the continuum, Handbook of set theory, Vols. 1, 2, 3, 395-489, Springer, Dordrecht, 2010. 
[2] A. Blass, S. Shelah, Ultrafilters with small generating sets, Israel Journal of Mathematics, 65 (1989), no. 3, 257-271 
[3] J. Brendle, V. Fischer, Y. Khomskii, Definable independent families, Transactions of the American Mathematical Society, to appear.
[4] V. Fischer, S. Friedman, D. Mejía, D. Montoya, Coherent systems of finite support iterations, Journal of Symbolic Logic, 83 (2018), no. 1, 208-236. 
[5] V. Fischer, D. Mejía, Splitting, bounding and almost disjointness can be quite different, Canadian Journal of Mathematics, 69 (2017), no. 3, 502-531.
[6] V. Fischer, D. Montoya, Ideals of independence, Archive for Mathematical Logic, to appear. 
[7] V. Fischer, S. Shelah, The spectrum of independence, Archive for Mathematical Logic, to appear.
[8] S. Shelah, On cardinal invariants of the continuum, Axiomatic set theory (Boulder, Colo., 1983), 183-207, Contemp. Math., 31, Amer. Math. Soc., Providence, RI, 1984. 
[9] S. Shelah, Two cardinal invariants of the continuum (mathfrak{d} < mathfrak{a}) and FS linearly ordered iterated forcing, Acta Mathematica, 192 (2004), no. 2, 187-223.