June 4th revolution in Ghana

Mabo Day in Australia on June 3 (It commemorates Eddie Koiki Mabo (c. 29 June 1936–21 January 1992) a Torres Strait Islander whose campaign for Indigenous land rights led to a landmark decision of the High Court of Australia that, on 3 June 1992, overturned the legal fiction of terra nullius which had characterised Australian law with regards to land and title since the voyage of James Cook in 1770);

Birthday of Carl Gustaf Emil Mannerheim & Flag Day celebration of the Finnish Defence Forces on June 4 (Finland);

Tiananmen Square Protests of 1989 Memorial Day on June 4 (International);

Memorial Day in South Korea on June 6;

Normandy landings of the Allied Expeditionary Forces on June 6 (D-Day), a.k.a. Operation Neptune, part of Operation Overlord (1944);

Battle of Arica Day on June 7 (Arica y Parinacota Region, Chile);

Sette Giugno in Malta on June 7;

Union Dissolution Day on June 7 (Norway. The Union Dissolution Day, observed in Norway on 7 June (though not a public holiday), is marked in remembrance of the Norwegian parliament's 1905 declaration of dissolution of the union with Sweden, a personal union which had existed since 1814)