Give me a six-word summary of the korean war.

Both Sides Declare Victory, Neither Won.

Explanation:

Technically, depending on which viewpoint you look at, both sides declared victory:

Communism: Communism declared victory (though it was a defeat on their part, because they did not complete their objective), because they were able to keep U.N. at bay from taking all of Korea.

Democracy: The US & U.N. declared victory (which was more of a victory, because they completed their primary objective) because they were able to stench Communist growth in Korea, as well as retake most of South Korea from Communist control.

Neither won: As we can see today, the Korean Peninsula is still split into two states, the North being a totalitarian state with China and Russia backing her up, while the South being a democratic-leaning state, backed by the US & U.N.

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The topic of the essay:  a. the purpose of government.

I won't write your essay for you, but as a Teacher can provide some advice as you prepare to write.  Lincoln's assertion is that the proper purpose of government is do for the people of a nation or community what individuals could not do very well if they each tried to do it themselves.  You might think of examples like infrastructure -- road building, ports, public utilities -- as matters that need community to come together rather than "every man for himself."  Similarly, matters of public safety like police protection in cities or national protection through a military are better done by a government working as a cooperative entity of the citizenry, rather than expecting each individual to provide all of his own protection and security.  There could be many other examples (including a contested one, such as health care) that could be used to give support in agreement with Lincoln's argument.