Explain what DuBois believes is the problem with the 20th-century. What does he mean by "the color-line?"

In W.E.B. DuBois's seminal work, The Souls of Black Folk, the problem of the twentieth-century is, indeed, that of the "color-line."

The phrase "the color-line" first appeared in an article called "The Color Line" written by Frederick Douglass for the North American Review in 1881. It became better-known when DuBois used the phrase to expand its relevance for the issues confronting black Americans in the twentieth-century.

The color-line is a direct reference to the segregation that existed in the United States -- both the legal, de jure, segregation in the South, and the de facto, or "in effect," racism in the North and the West. Segregation made it so that blacks and whites lived in such disparate worlds that their lives generally did not look at all the same. Though black people were very often aware the rights and privileges that were kept from them, particularly since working-class blacks were very often employed in white homes as domestic laborers, many white people were blind -- often willfully -- to the inferior conditions in which black people existed.

The color-line created inequalities in education, housing, and access to employment. It made it nearly impossible for black and white people to have any amity between them. Existentially, it worked to convince black people of their presumed inferiority. 

However, DuBois was concerned not only with the conditions in which black people existed in the United States, but also the lives of darker peoples in other countries, particularly the colonies in Africa, Asia, and Oceania. The subjugation of these peoples was justified by pseudo-scientific racial theories which attempted to give "the color-line" a scientific basis.