Did or did not Saul know who David was?
1 Samuel 16:19-23 and 1 Samuel 17:55-58
    

1. Yes, Saul knew David (1 Samuel 16:19-23) - "So Saul sent messengers to Jesse, and said, "Send me your son David who is with the flock." ²⁰And Jesse took a donkey loaded with bread and a jug of wine and a young goat, and sent them to Saul by David his son. ²¹Then David came to Saul and attended him, and Saul loved him greatly; and he became his armor bearer. ²²And Saul sent to Jesse, saying, "Let David now stand before me; for he has found favor in my sight." ²³So it came about whenever the evil spirit from God came to Saul, David would take the harp and play it with his hand; and Saul would be refreshed and be well, and the evil spirit would depart from him," (NASB).
2. Saul asked "Whose son are you?" (1 Samuel 17:55-58)  - "Now when Saul saw David going out against the Philistine, he said to Abner the commander of the army, "Abner, whose son is this young man?" And Abner said, "By your life, O king, I do not know." ⁵⁶And the king said, "You inquire whose son the youth is." ⁵⁷So when David returned from killing the Philistine, Abner took him and brought him before Saul with the Philistine’s head in his hand. ⁵⁸And Saul said to him, "Whose son are you, young man?" And David answered, "I am the son of your servant Jesse the Bethlehemite," (NASB).

     Several possible explanations exist to explain this problem ranging from textual error to the method of the ancient writers to write about themes and then come back to fill in the gaps later.  I think the best explanation can be found by looking at the context.
     There is no contradiction.  Saul was imply asking David whose son he was.  He knew David but probably didn't know who was David's father.  Since David has just saved Israel, Saul wanted to know who his father was, probably to show the father proper respect for his son David.
     In 1 Samuel 16, the Spirit of the Lord had left Saul (16:14) and an evil spirit came and afflicted Saul (16:23).  David came and then played for Saul to sooth him.  Chapter 17 begins the well known story of David and Goliath with no mention of how much time passes between David playing the harp and Goliath's challenge.  It may very well have been many months or even years.  Nevertheless, David was the youngest of the sons of Jesse (17:14), who was a youth (17:33), and who tended the flocks (17:15).  David is then known as a young musician and a sheep herder, not a warrior as were his three oldest brothers (17:13-14).  Saul and David have conversations about David doing battle with Goliath and Saul offers David his armor (17:38).  David refuses the armor and goes out to kill Goliath. Saul then asks Abner, "Whose son is this young man?" And Abner said, "By your life, O king, I do not know," (17:55). In verse 58 Saul says, "Whose son are you, young man?" And David answered, "I am the son of your servant Jesse the Bethlehemite."