Abstract
We are given $n$ balls and an unknown coloring of them with two colors. Our goal is to find a ball that belongs to the larger color class, or show that the color classes have the same size. We can ask sets of $k$ balls as queries, and the problem has different variants, according to what the answers to the queries can be. These questions has attracted several researchers, but the focus of most research was the adaptive version, where queries are decided sequentially, after learning the answer to the previous query. Here we study the non-adaptive version, where all the queries have to be asked at the same time.
Highlights
A widely studied problem in combinatorial search theory is the so-called Majority Problem
We would like to find a ball of the majority color or show that there is no majority color by asking subsets of [n], that we call queries
We would like to determine the minimum number of queries needed in the worst case with an optimal strategy if all the queries are fixed at the beginning
Summary
A widely studied problem in combinatorial search theory is the so-called Majority Problem. For k ≥ 1, let us denote by m(k) the cardinality of the edge set of a smallest k-uniform hypergraph that does not have Property B This parameter is widely studied, the best lower bound on m(k) we are aware of is Ω(2k k/ log k) due to Radhakrishnan and Srinivasan [13]. Let us denote by d(k) the cardinality of the edge set of a smallest k-uniform hypergraph that does not have Property C. We consider d(k, n), which is the cardinality of the edge set of a smallest k-uniform hypergraph with n vertices that does not have Property C.
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