기초과학연구원 이산수학그룹
On January 13, 2026, Ferdinand Ihringer from the Southern University of Science and Technology, China gave a talk on low-degree Boolean functions and appications of vector space Ramsey numbers at the Discrete Math Seminar. The title of his talk was “Boolean Functions Analysis in the Grassmann Graph“.
Boolean function analysis for the hypercube $\{ 0, 1 \}^n$ is a well-developed field and has many famous results such as the FKN Theorem or Nisan-Szegedy Theorem. One easy example is the classification of Boolean degree $1$ functions: If $f$ is a real, $n$-variate affine function which is Boolean on the $n$-dimensional hypercube (that is, $f(x) \in \{ 0, 1 \}$ for $x \in \{ 0, 1 \}^n$), then $f(x) = 0$, $f(x) = 1$, $f(x) = x_i$ or $f(x) = 1 – x_i$. The same classification (essentially) holds if we restrict $\{ 0, 1\}^n$ to elements with Hamming weight $k$ if $n-k, k \geq 2$. If we replace $k$-sets of $\{ 1, \ldots, n \}$ by $k$-spaces in $V(n, q)$, the $n$-dimensional vector space over the field with $q$ elements, then suddenly even the simple question of classifying Boolean degree $1$ functions, here traditionally known as Cameron-Liebler classes, becomes seemingly hard to solve.
We will discuss some results on low-degree Boolean functions in the vector space setting. Most notably, we will discuss how vector space Ramsey numbers, so extremal combinatorics, can be utilized in this finite geometrical setting.