Graph Theory: I understanding the corollaries

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Terrell
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please check the attached photo
 

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Terrell said:
please check the attached photo

What is your question?
 
Math_QED said:
What is your question?
when is the intersection number less than the number of edges?

also a clarification i need, in Si s.t. Si is subset of S, does it contain v_i together with edges incident to it? so meaning it is a set containing a vertex and edges?
 
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