Abstract This paper will consider combinatorial properties related to coding a cardinal by its bounded subsets. These properties have traditionally been studied in the context of very large cardinals and variations of these properties either reach the level of Kunen inconsistency or are very close to it. Within the descriptive set‐theoretic framework with determinacy or partition properties, these combinatorial properties are quite robust and have numerous natural examples. Let be a cardinal, , and . is the set of all subsets of of ordertype which are bounded below . is the set of all subsets of of ordertype less than bounded below . The following will be shown which answer or address several questions of Ben‐Neria and Garti. Let be the club filter on . Assume and . For any function , there is an with so that . Let be the ‐club filter on . If and hold, then for any and any function , there is an so that . Let be the supremum of the ordinal onto which surjects. For any cardinal with , there is a function so that for all , . Assume and . For any uniform countably complete filter on , there is a function so that for all , . Assume . Let be the supremum of the projective ordinals. For any and , there is an with so that . There is also a uniform filter on so that for all and function , there is an so that .
William Chan (Sun,) studied this question.
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