On two intersecting set systems and k-continuous boolean functions

@article{Tuza1987OnTI,
title={On two intersecting set systems and k-continuous boolean functions},
author={Zsolt Tuza},
journal={Discrete Applied Mathematics},
year={1987},
volume={16},
pages={183-185}
}

(A Boolean function f : { 0 , 1 }n~{0 , 1} is called k-continuous if for every x = (Xl, . . . , xn) E {0, 1 }n there exists a sequence 1 _ il < --" < ik < n such that f ( x ) = f ( y ) for all Y=(Yl, . . . ,Yn)e {0, 1} n with yil=xi,, ...,yik=Xik.) There was a gap, however, between the orders of magnitude of the upper and lower bound for m(k, k). Consider now two set systems ~¢= {A 1, ... ,An} and ~ = {B 1, ... ,Bn} of the same cardinality and with the following properties.