Forbidden vector-valued intersection

Given vectors $V = (v_i: i \in [n]) \in R^D$, we define the $V$-intersection of $A,B \subset [n]$ to be the vector $\sum_{i \in A \cap B} v_i$. In this talk, I will discuss a new, essentially optimal, supersaturation theorem for $V$-intersections, which can be roughly stated as saying that any large family of sets contains many pairs $(A,B)$ with $V$-intersection $w$, for a wide range of $V$ and $w$. A famous theorem of Frankl and Rödl corresponds to the case $D=1$ and all $v_i=1$ of our theorem. The case $D=2$ and $v_i=(1,i)$ solves a conjecture of Kalai.

Joint work with Peter Keevash.