What if a police officer arrests me in my home can they search my house?

The police can search the area in which you were arrested, anywhere beyond that is an invalid search.

If you believe a search is being conducted unlawfully, tell the officers that you do not consent to a search, and wish to remain silent. Stay calm and document the situation as best as possible with a specific details in order to provide to a lawyer if necessary.

The area near where you are arrested can be searched without a warrant, but your entire house cannot be searched unless there is a warrant that describes in detail the places to be searched and the people or things to be seized.