Find the maximum number of subsets that satisfy this trait
~ Each Subset has 4 element
~ Each two subset share 2 elements in common
~ There can't be more than 1 number that are included in all subsets
I wanted to transform this into a projective geometry approach by taking each subset as a cyclic quadrilateral and each intersection as a 2 of the point in each cyclic Quadri
Link That Motivates (https://brilliant.org/wiki/projective-geometry/)
could someone here give a hint or help me solve this with using any method
I haven't be able to visualize the reference you give because one has to be a member of such and such for that.
Nevertheless, one can exhibit a solution complying with the different constraints ; take the set of 4 elements subsets of $\{0,1,2,3,4,5,6\}$ which are :
$$\begin{cases} \{0,1,2,6\}\\ \{0,1,4,5\}\\ \{0,2,3,5\}\\ \{0,3,4,6\}\\ \end{cases}$$ represented here (please note that vertices 5 and 6 are represented twice):
Is it a maximal solution ? In terms of the number of subsets (as you ask) ? In terms of the total number of elements involved ? It remains an open question.