## Variants of Ramsey theorem

### Task number: 4016

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.