Burr, Erdős, and Lovász, in 1976, introduced the study of the smallest minimum degree s(r,k) of a graph G such that any r-colouring of the edges of G contains a monochromatic clique of size k, whereas no proper subgraph of G has this property. Burr, Erdős, and Lovász were able to show the rather surprising exact result, that if r=2, then s(2,k) = (k-1)^2. The behaviour of this function is still not so well understood for more than 2 colours. In 2016, Fox, Grinshpun, Liebenau, Person, and Szabó showed that for r>2, s(r,k) is at most 8(k-1)^6 r^3. The speaker, together with Anurag Bishnoi and Thomas Lesgourgues, have recently used finite geometry to improve this bound.
