Give two possible themes for "Battle Royal." Give examples from the story to support your answer.

"Battle Royal" is chapter one of Ralph Ellison's novel Invisible Man.  Many of the themes that run throughout the book are present in chapter one.  The two themes that I will discuss from this chapter are oppression and sexual objectification. 
Let's look at oppression in "Battle Royal" first.  What I find interesting about this chapter is that the events of the chapter not only show that the narrator is oppressed because of his skin color, but the chapter also shows that women are oppressed.   
The narrator is asked to give a speech at a gathering of the town's leading white citizens, and the narrator is both proud and excited for the opportunity.  Unfortunately, the narrator is forced to take part in the evening's "entertainment."

Everyone praised me and I was invited to give the speech at a gathering of the town's leading white citizens. It was a triumph for the whole community.
It was in the main ballroom of the leading hotel. When I got there I discovered that it was on the occasion of a smoker, and I was told that since I was to be there anyway I might as well take part in the battle royal to be fought by some of my schoolmates as part of the entertainment.

The entertainment is the "battle royal."  It is not entertainment, but instead intentional humiliation and oppression of both black boys and white women.  The black boys are forced to fight each other while blindfolded.  The white men then shout racial slurs and obscenities at the boys to encourage them to fight harder. 

"Let me at those black sonsabitches!" someone yelled.
"No, Jackson, no!" another voice yelled. "Here, somebody, help me hold Jack."
"I want to get at that ginger-colored nigger. Tear him limb from limb," the first voice yelled. 

After the fighting is over, the boys are then suckered into diving down for coins on an electrified rug.

I heard, "These niggers look like they're about to pray!"
Then, "Ready", the man said. "Go!"
I lunged for a yellow coin lying on the blue design of the carpet, touching it and sending a surprised shriek to join those around me. I tried frantically to remove my hand but could not let go. A hot, violent force tore through my body, shaking me like a wet rat. The rug was electrified. . . . The men roared above us as we struggled.  

The narrator is forced to do these awful things simply because he is black.  He learns that the white men do not view him as a fellow, "equal" man.  

"You sure that about 'equality' was a mistake?"

Whether they do not view him as a man at all or simply not an equal man is debatable; however, it ultimately doesn't matter to the narrator.  He realizes that the men do not see him as a man.  That's why the first paragraph tells readers that the narrator had to learn about his invisibility.  

But first I had to discover that I am an invisible man!

Women are also oppressed and abused in this chapter.  Before the boys are forced to fight each other, they are forced to watch a naked white woman dance in front of them.  The boys are ordered to watch her dance, and the men enjoy watching the woman and the discomfort of the young boys.  The sequence is incredibly demeaning to the woman because she is being forced to present herself as a sexual object to the boys.  What makes it worse, is that the white men view her this way too.  That is made clear when some of the white men can't control themselves anymore and begin groping her.  Some of the men eventually grab her and throw her on the ground.  

Chairs went crashing, drinks were spilt, as they ran laughing and howling after her. They caught her just as she reached a door, raised her from the floor, and tossed her as college boys are tossed at a hazing, and above her red, fixed-smiling lips I saw the terror and disgust in her eyes, almost like my own terror and that which I saw in some of the other boys. As I watched, they tossed her twice and her soft breasts seemed to flatten against the air and her legs flung wildly as she spun. Some of the more sober ones helped her to escape.  

The woman is oppressed in two ways.  She is oppressed for being a woman, and she is oppressed further because she is only valued as a sexual object.  That's an incredibly demeaning and dehumanizing way of looking at a woman.  Because she is an object, she is just as powerless as the boys. Simple grammar teaches that subjects act and do, and objects are acted upon by the subject.  The narrator describes this concept to readers when he describes what he would like to do to the naked woman. 

I wanted at one and the same time to run from the room, to sink through the floor, or go to her and cover her from my eyes and the eyes of the others with my body; to feel the soft thighs, to caress her and destroy her, to love her and to murder her, to hide from her, and yet to stroke where below the small American flag tattooed upon her belly her thighs formed a capital V. 

The woman is something that the narrator wants to possess.  In this regard, he isn't any different than the lecherous men that attempt to grope and subdue the woman.  By the end of the chapter, a reader has been given a very vivid picture of what sexual objectification and various types of oppression look like.