Choosing between incompatible ideals

Suppose I and J are proper ideals on some set X. We say that incompatible if I∪J does not generate a ideal. Equivalently, there is A⊆X such A∈I X∖A∈J. If B⊆X either in I∖J or J∖I, then we B chooses between J. consider the following Ramsey-theoretic problem: Given several pairs (I1,J1),(I2,J2),…,(Ik,Jk) of X, find as many these possible. The main theorem for every n∈N, I(n)∈N given at least I(n) any choosing n them. This proved two steps. first step to identify (purely finitary) problem extremal combinatorics, show our concerning equivalent this combinatorial problem. second analyze order number described above exists, put bounds it. 12nlog2n−O(n)<I(n)<nlnn+O(n). upper bound by considering different but closely related involving hypergraphs, which may be independent interest. also investigate applications conditionally convergent series.