Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Yurii Khomskii observed that \({{\mathrm{cof}}}(l^0)>\mathfrak {c}\) assuming \(\mathfrak {b}=\mathfrak {c}\) and he asked whether the inequality \({{\mathrm{cof}}}(l^0)>\mathfrak {c}\) is provable in ZFC.  find several conditions that imply some variants of this inequality for tree ideals. Applying a recent result of Brendle, Khomskii, and Wohofsky  show that \(l^0\) satisfies some of these conditions and consequently, \({{\mathrm{cof}}}(l^0)=\mathfrak {d}(^\mathfrak {c}l^0)\ge \mathfrak {d}(^\mathfrak {c}\mathfrak {c})>\mathfrak {c}\).  also prove that if the cellularity of a Boolean algebra B is hereditarily \(\ge \kappa \), then every \(\kappa \)-sequence in \(B^+\) has a \(\kappa \)-subsequence with a disjoint refinement.
['Miroslav Repický']
I am writing a research paper on a idea Cofinality of the laver ideal
[{"from": "human", "value": "I am writing a research paper on a idea Cofinality of the laver ideal"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Yurii Khomskii observed that \\({{\\mathrm{cof}}}(l^0)>\\mathfrak {c}\\) assuming \\(\\mathfrak {b}=\\mathfrak {c}\\) and he asked whether the inequality \\({{\\mathrm{cof}}}(l^0)>\\mathfrak {c}\\) is provable in ZFC.  find several conditions that imply some variants of this inequality for tree ideals. Applying a recent result of Brendle, Khomskii, and Wohofsky  show that \\(l^0\\) satisfies some of these conditions and consequently, \\({{\\mathrm{cof}}}(l^0)=\\mathfrak {d}(^\\mathfrak {c}l^0)\\ge \\mathfrak {d}(^\\mathfrak {c}\\mathfrak {c})>\\mathfrak {c}\\).  also prove that if the cellularity of a Boolean algebra B is hereditarily \\(\\ge \\kappa \\), then every \\(\\kappa \\)-sequence in \\(B^+\\) has a \\(\\kappa \\)-subsequence with a disjoint refinement."}]