Nasty Finite Combinatorics

Please comment your solutions, questions and remarks..

I was blamed for having too easy Wednesday problems. So beware! The level is coming up!

Suppose that K is a set of n elements, [tex]n\in\mathbb{N}[/tex]. Suppose that [tex]K_1,\dots,K_n,K_{n+1}[/tex] are subsets of K. Show that there exist [tex]I\textrm{ and }J\subset \{1,\dots,n,n+1\}[/tex] such that

[tex]I\cap J =\varnothing[/tex]

[tex]I\cup J\ne \varnothing[/tex]

and

[tex]\displaystyle \bigcup_{i\in I}K_i =\bigcup_{j\in J}K_j[/tex]

4/5

Hint below printed in white, so that if you are using a common OS with common settings, you have to drag with your mouse over the text in order to see it:

Use something else than finite combinatorics.

About Vadim Kulikov

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