MathDB

A division of a group of people into various groups is called

k

-regular if the number of groups is less or equal to

k

and two people that know each other are in different groups. Let

A

B

, and

C

groups of people such that there are is no person in

A

and no person in

B

that know each other. Suppose that the group

A \cup C

has an

a

-regular division and the group

B \cup C

has a

b

-regular division. For each

a

and

b

, determine the least possible value of

k

for which it is guaranteed that the group

A \cup B \cup C

has a

k

-regular division.

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