The book of Philemon demonstrates the gospel in action. Because believers in Jesus are equal partners in the new humanity, Paul asks Philemon to forgive his runaway slave Onesimus.
Paul urges Philemon to forgive Onesimus and accept him back as an equal. Because they are both believers, God's grace and healing mercy have made them partners under the new humanity Jesus' established.
While Paul doesn't mention Jesus' death and resurrection directly in this book, he offers to take on any punishment Onesimus deserves, a demonstration of what Jesus did on the cross. He encourages Philemon to do the same as he reconciles to Onesimus before God.