ALLINGE, Denmark — For four days a year in June, Danes like to pretend there are no boundaries between them.

They gather for Folkemodet, a political festival where the prime minister, chief executives of top companies and other respected leaders remove their ties and heels and mingle with members of the public on the remote island of Bornholm in the Baltic Sea.

Here, the young, the old and the in-betweens have frank debates about the state of their democracy, holding forth over hot dogs and beer, ducking in and out of tents to hear speeches on issues both profound and personal, and, perhaps, helping to find solutions to problems in their society.

Earlier this month, 110,000 people descended on the postcard-perfect island town of Allinge, with its two-story houses with red tile roofs, stone hedges and gardens with fruit trees and roses.