A battle of the sexes is any competition between females and males, or the persistent rivalry or opposition between males and females.
The Battle of Lemberg was a battle between West Ukrainian People's Republic and local Polish people. This battle was or the control of the city of L'viv.
The Battle of Actium was the definitive clash of two Romans for supreme power. Although the war was nominally against Cleopatra, it was actually a civil war between the forces of Octavian and the forces of Antony.
Battle of Gallipoli of 1915.
Battle, Contest, race, rivalry, strife, striving, struggle, tug of war, war,warfare
Without a context, this cannot be clearly determined. However, an "arena" is an area of battle or conflict, so the term is being used metaphorically to indicate: 1) a rivalry, as between two opposing individuals 2) a rivalry between groups for social dominance 3) a contest of some type, such as which cheerleaders are the most vibrantly dressed 4) an actual fight, probably a food fight.
The Battle of Fallen Timbers was a battle between the United States and the Western Confederacy, which included Great Britain. The result of the battle was a victory for the United States, which gave the U.S. control over the Northwest Territory.
Battle of Vicksburg
On that date, King Edward ("The Confessor") of England died leaving no clear heir. This ignited the three-way rivalry between William the conqueror, Harold the king's brother, Harald the King of Norway.
The battle of Britain.
The Battle of Vicksburg gave the Union control of the Mississippi River.
the hostilities between Allied convoys and German U-boats for control of supplies lines.