Find the minimum number
of subsets in a separating
family for a set of elements, where a separating
family is a set of subsets
in which each pair of adjacent elements is found separated, each in one of two disjoint subsets. For example, the 26 letters of the
alphabet can be separated by a family of nine:
The problem was posed by Katona (1973) and solved by C. Mao-Cheng in 1982,
where
is the ceiling function. is nondecreasing, and the values for , 2, ... are 0, 2, 3, 4, 5, 5, 6, 6, 6, 7, ... (OEIS A007600).
The values at which
increases are 1, 2, 3, 4, 5, 7, 10, 13, 19, 28, 37, ... (OEIS A007601),
so ,
as illustrated in the preceding example.
Honsberger, R. "Cai Mao-Cheng's Solution to Katona's Problem on Families of Separating Subsets." Ch. 18 in Mathematical
Gems III. Washington, DC: Math. Assoc. Amer., pp. 224-239, 1985.Katona,
G. O. H. "Combinatorial Search Problem." In A
Survey of Combinatorial Theory (Ed. J. N. Srivasta, F. Harary, C. R.
Rao, G.-C. Rota, and S. S. Shrikhande). Amsterdam, Netherlands: North-Holland,
pp. 285-308, 1973.Sloane, N. J. A. Sequences A007600/M0456
and A007601/M0525 in "The On-Line Encyclopedia
of Integer Sequences."
of subsets in a separating
family for a set of 
elements, where a separating
family is a set of subsets
in which each pair of adjacent elements is found separated, each in one of two disjoint subsets. For example, the 26 letters of the
alphabet can be separated by a family of nine:
![f(n)=min{2p+3[log_3(n/(2^p))]:p=0,1,2},](/images/equations/KatonasProblem/NumberedEquation2.svg)
![[x]](/images/equations/KatonasProblem/Inline3.svg)
is the ceiling function. 
is nondecreasing, and the values for 
, 2, ... are 0, 2, 3, 4, 5, 5, 6, 6, 6, 7, ... (OEIS A007600).
The values at which 
increases are 1, 2, 3, 4, 5, 7, 10, 13, 19, 28, 37, ... (OEIS A007601),
so 
,
as illustrated in the preceding example.
