Query complexity of Boolean functions on slices - Byramji, Farzan (Discrete mathematics, 2024)
The kth slice ([n]k) of the Boolean cube {0,1}n is the set of all n-bit strings with Hamming weight k. We study the deterministic query complexity of Boolean functions on slices of the Boolean cube. We show that there exists a function on the balanced slice ([n]n/2) requiring n−O(log⁡log⁡n) queries. We observe that there is an explicit function on the balanced slice requiring n−O(log⁡n) queries based on independent sets in Johnson graphs. We also consider the maximum query complexity on constant-weight slices and how it relates to Ramsey theorems.