People say I'm sleeping with someone else. Can I sue him and send text messages on my cell phone?

Yes, this is slander. There is no provision for the crime of libel in our country's laws. If the circumstances of insult and libel are serious, it may constitute the crime of libel. Defamation refers to deliberate fabrication and dissemination of fictional facts, which is enough to degrade others' personality and damage their reputation, and the circumstances are serious. Subjectively, it must be intentional. The actor knows that he is spreading false facts that can damage the reputation of others, knows that his actions will lead to harmful results that will damage the reputation of others, and hopes that such results will happen. The actor's purpose is to discredit others. If the actor misunderstands false facts as true facts and spreads them, or spreads some false facts for the purpose of damaging the reputation of others, it does not constitute libel. Judging from the situation you described, first, the other party just sent you a short message, which was not publicly distributed in the society; Secondly, the circumstances did not reach the seriousness stipulated by law. So it may be difficult for you to sue each other for libel.