Question:  Between the Battle of Pusan Perimeter and The Battle of Luzon which had the most combatants?

Context paragraphs:
15th Division (North Korea): The 15th Infantry Division was a military formation of the Korean People's Army during the 20th Century.  The division fought in the 1950 Korean War; it took part in the North Korean advance from Seoul to Taejon, and fought in the Battle of Pusan Perimeter.  The 15th Division fought along the eastern coast, above Pusan, eventually being annihilated.

Battle of Pusan Perimeter: The Battle of Pusan Perimeter was a large-scale battle between United Nations and North Korean forces lasting from August 4 to September 18, 1950.  It was one of the first major engagements of the Korean War.  An army of 140,000 UN troops, having been pushed to the brink of defeat, were rallied to make a final stand against the invading North Korean army, 98,000 men strong.

Battle of Luzon: The Battle of Luzon, fought 9 January – 15 August 1945, was a land battle of the Pacific Theater of Operations of World War II by the Allied forces of the U.S., its colony the Philippines, and allies against forces of the Empire of Japan.  The battle resulted in a U.S. and Filipino victory.  The Allies had taken control of all strategically and economically important locations of Luzon by March 1945, although pockets of Japanese resistance held out in the mountains until the unconditional surrender of Japan.  While not the highest in U.S. casualties, it is the highest net casualty battle U.S. forces fought in World War II, with 192,000 to 205,000 Japanese combatants dead (mostly from starvation and disease), 10,000 American combatants killed, and between 120,000 to 140,000 Filipino civilians and combatants killed.

Battle of Pusan Perimeter logistics: Logistics in the Battle of Pusan Perimeter (August 4 – September 15, 1950) during the Korean War played a decisive role in the battle.  Efficient logistics, the management of personnel and materiel, supported United Nations (UN) supply lines while the North Koreans' routes of supply were steadily reduced and cut off.  UN logistics improved throughout the Battle of Inchon and the defeat of the North Korean army at Pusan.

Battle of Ka-san: The Battle of Ka-san was an engagement between United Nations (UN) and North Korean forces early in the Korean War from September 1 to September 15, 1950, in the vicinity of Ka-san in South Korea.  It was a part of the Battle of Pusan Perimeter, and was one of several large engagements fought simultaneously.  The battle ended in a victory for the United Nations after large numbers of United States (US) and Republic of Korea (ROK) troops repelled a strong North Korean attack.

Battle of Battle Mountain: The Battle of Battle Mountain was an engagement between United Nations (UN) and North Korean (NK) forces early in the Korean War from August 15 to September 19, 1950, on and around the Sobuk-san mountain area in South Korea.  It was one of several large engagements fought simultaneously during the Battle of Pusan Perimeter.  The battle ended in a victory for the UN after large numbers of United States (US) and Republic of Korea (ROK) troops were able to prevent a North Korean division from capturing the mountain area.

First Battle of Naktong Bulge: The First Battle of Naktong Bulge was an engagement between United States and North Korean forces early in the Korean War from August 5–19, 1950 in the vicinity of Yongsan (Yeongsan, Changnyeong county) and the Naktong River in South Korea.  It was a part of the Battle of Pusan Perimeter, and was one of several large engagements fought simultaneously.  The battle ended in a victory for the United States after large numbers of US reinforcements destroyed an attacking North Korean division.

Battle of P'ohang-dong: The Battle of P'ohang-dong was an engagement between the United Nations and North Korean forces early in the Korean War, with fighting continuing from 5–20 August 1950 around the town of P'ohang-dong, South Korea.  It was a part of the Battle of Pusan Perimeter, and was one of several large engagements fought simultaneously.  The battle ended in a victory for the United Nations after their forces were able to drive off an attempted offensive by three North Korean divisions in the mountainous eastern coast of the country.

Battle of Taegu: The Battle of Taegu was an engagement between UN and North Korean forces early in the Korean War, with fighting continuing from August 5–20, 1950 around the city of Taegu, South Korea.  It was a part of the Battle of Pusan Perimeter, and was one of several large engagements fought simultaneously.  The battle ended in a victory for the United Nations after their forces were able to drive off an offensive by North Korean divisions attempting to cross the Naktong River and assault the city.

Battle of the Bowling Alley: In the Battle of the Bowling Alley (August 12–25, 1950), United Nations (UN) forces defeated North Korean forces early in the Korean War near the city of Taegu, South Korea.  The battle took place in a narrow valley, dubbed the "Bowling Alley", which was north of Taegu.  It followed a week of fighting between the Korean People's Army (KPA) 13th Division and the Republic of Korea Army's (ROK) 1st Division along the latter's last defensible line in the hills north of the city.  Reinforcements, including the US Army's 27th and 23rd Infantry Regiments were committed to bolster the South Koreans' defenses.  This battle and several others were smaller engagements of the Battle of Pusan Perimeter.

Supporting paragraphs: Battle of Luzon Battle of Pusan Perimeter
Answer: The Battle of Luzon