I am trying to prove or disprove the following statement: Let $n>1$ be a positive integer. Then there exists a graph $G$ of size 4n-1 such that if the edges of $G$ are colored red or blue, no matter in which way, $G$ definitely contains a monochromatic $K_{2,n}$.
I tried to check a few cases in the hope of discovering a counter-example. For n=2 $G$ has to have size 7. The graph certainly is of the form "Square + 3 edges". Moreover, it should have the property that if among the 7 edges any 3 are deleted the remaining graph is a square. I couldnt construct any such graph. Is there any justification why such a graph cant exist, thereby negating the statement?