A q-analogue of the binomial coefficients and applications to off-diagonal Ramsey numbers

q-analogues of quantities in mathematics involve perturbations of classical quantities using the parameter q, and revert to the original quantities when q goes 1. A notable example is the q-analogues of binomial coefficients, denoted by {n \choose k}_q, which give the number of k-dimensional subspaces in F_q^n. When q goes to 1, this reverts to the binomial coefficients, which measure the number of k-sets in [n]. 

In this talk, we consider one more structure in F_q^n, which is the Euclidean quadratic form: Euc_n =x_1^2 + x_2^2 + ... x_n^2. The main goal of this talk is to define the Euclidean-analogues of the binomial coefficients and explore related combinatorics. We show that the number of quadratic subspaces of Euclidean type in (F_q^n, Euc_n) can be described as the form of the analogue of binomial coefficients. This construction establishes a parallel to the q-binomial coefficients associated with orthogonal groups. Finally, we will apply this to investigate the lower bound of off-diagonal Ramsey numbers.