Probability and random variable

Question

Consider the sets $A_1,  A_2,  A_3  \dots  A_m$ each a subset of size $k$ of $\{1,2,3, \dots , n\}$.  If in a 2-colouring of $\{1,2,3, \dots , n\}$ no set $A_i$ is monochromatic, then show that $ m < 2^{k-1}$.

Comments

https://ktiml.mff.cuni.cz/~fink/teaching/data_structures_I/matousek-vondrak-prob-ln.pdf

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chk 2.2.4 proving

is it right one?

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In the above problem assume m=3,n=5 and k=2

So, here the sets are A₁,A_2,A₃ each is a subset of size 2 of {1,2,3,4,5}

Now, we colour the nos. 1,2,3,4 and 5

1 → BLUE

2 → RED

3 → BLUE

4 → RED

5 → BLUE

Now, we are forming the sets A₁,A₂,A₃

A₁ = { 1,2 }

A₂ = { 3,4}

A₃ = { 5,4 }

Check that none of the above sets are monochromatic and each of them is a subset of { 1,2,3,4,5 }

Now, here m= 3 and k= 2

Thus m> 2^k-1

which does not satisfy the condition which we have to find in the question m< 2^k-1

I think something is missing or the conditions given are wrong.

Debashis can you please check the problem or tell me if I am wrong somewhere.