Burke was, along with Alderman Edward Vrdolyak, a leader of the "Vrdolyak 29", a city council majority voting block, which included 28 white and one Puerto Rican aldermen, who opposed the agenda of the newly elected Mayor Harold Washington, Chicago's first black mayor, during Washington's first three years as mayor, 1983–1986, a period referred to as Council Wars. Vrdolyak, a Burke mentor, was chairman of the Cook County Democratic Party. Vrdolyak forged the alliance by expanding the number of city council committees to 29 and negotiating 29 committee chairmanship assignments. On May 2, 1983, during the first city council meeting of the Washington administration, the mayor and some aldermen left the meeting, Vrdolyak was President "pro tempore" of the city council (chair of city council meetings when the mayor is not present) so he continued the meeting. Burke was chosen to chair the powerful Finance Committee.