Explore the fascinating world of mathematical connections in this 52-minute lecture by Dr. Colva Roney-Dougal, Senior Lecturer in Pure Mathematics at the University of St Andrews. Discover the intriguing question of how many party guests are needed to guarantee that at least five of them either all know each other or are mutual strangers. Delve into unexpected applications of this unsolved mathematical problem, from modeling flu epidemics to understanding galaxy formation. Delivered as part of the Cambridge Festival 2013 at the Isaac Newton Institute for Mathematical Sciences, this engaging talk titled "Party Hard! The Maths of Connections" offers a unique perspective on the intersection of mathematics and real-world phenomena.