Variants of Ramsey theorem

Decide which of the following statements are true:

• Variant

For every positive integer $$n$$ there exists a positive integer $$N$$ such that if the vertices of the complete graph $$K_N$$ are colored with two colors, then in the considered graph exists a complete subgraph $$K_n$$ whose all vertices are colored with the same color.

• Variant

If we color a sufficiently large complete graph with loops with two colors (we color edges and loops), there is always a monocromatic complete subgraph with loops on $$n$$ vertices.

• Variant

For every positive integer $$n$$ there exists a positive integer $$N$$ such that for any graph $$G$$ on $$N$$ vertices holds: either $$G$$ contains $$K_{n,n}$$ as a subgraph or the complement of $$G$$ contains $$K_{n,n}$$ as a subgraph. Considered subgraph need not to be induced.

• Variant

For every positive integer $$n$$ there exists a positive integer $$N$$ such that for any graph $$G$$ on $$N$$ vertices holds: either $$G$$ contains $$K_{n,n}$$ as a subgraph or $$G$$ contains the complement of $$K_{n,n}$$ as a subgraph. Considered subgraph need not to be induced.