Sub-Ramsey Numbers for Arithmetic Progressions and Schur Triples

In this talk we exhibit new upper bounds for the order of ( k , g ) -cages when k − 1 is not a prime power and g ∈ { 6 , 8 , 12 } . As an application we obtain new upper bounds for the order of cages when g = 11 and g = 12 and k − 1 is not a prime power. We also confirm a conjecture of Fu, Huang and Rodger on the k-connectivity of ( k , g ) -cages for g = 12 , and for g = { 7 , 11 } when k − 1 is a prime power.