I am a new to graph theory. In the 4-th lecture given by our instructor Ramsey's theorem was introduced to us. Let $[n]=\{1,2, \cdots , n \}$. He has given the statement as follows $:$
Given positive integers $k,r,l$ ($l \geq k$), $\exists$ a positive integer $n=n(k,r,l)$ such that if the collection of $k$-subsets of $[n]$ is of $r$-coloured then $\exists$ an $l$-element subset $L$ of $[n]$ such that all the $k$-subsets of $L$ are monochromatic.
The generalized version of Ramsey's theorem is given as follows $:$
Given positive integers $l_1,l_2, \cdots , l_r$ such that for any $r$-coloring of the $k$-subsets $\exists$ some $1 \leq i \leq r$ such that $\exists$ an $l_i$-subset $L$ of $[n]$ such that all the $k$-subsets of $L$ are monochromatic.
But I can't find the similarity of the above theorem given by our instructor and the same given in wikipedia. What's going wrong in it?
Please help me in this regard.
Thank you very much.
The second statement is a generalization of the form of the theorem given on Wikipedia, which is the special case $k=2$. In the case $k=2$, the $2$-subsets of $[n]$ correspond to the edges in the complete undirected graph on $[n]$.