5 problems

The palindrome-or-two-palindromes trace conjecture. If $w$ has positive trace for every pair of real positive definite matrices $A$ and $B$, then $w$ is a palindrome or a product o…

Let $W(n,k,r)$ denote the maximum size of a family $\mathcal{A}\subset [n]^{(k)}$ whose trace on every $r$-subset of $[n]$ contains no maximal chain. The complete extremal-family c…

Let $W(n,k,r)$ be the maximum size of a family $\mathcal{A}\subset [n]^{(k)}$ whose trace on every $r$-subset of $[n]$ contains no maximal chain, and let $n_0(k,r)$ denote a thresh…

Let $[n]$ be the set $\{1,2,\ldots,n\}$, let $[n]^{(k)}$ denote the family of $k$-subsets of $[n]$, and let the trace of a family $\mathcal{A}$ on $X$ be … A maximal chain of an…

Let $G$ be a group, and let $K_{0}(\mathbb{Z}G)$ be the Grothendieck group of finitely generated projective $\mathbb{Z}G$-modules. Write $\epsilon$ for the augmentation trace and…