6 problems

Let $M$ be a $\{-1,1\}$ matrix. Call $M$ good if every matrix obtained by independently switching any subset of its diagonal entries is nonsingular. Let $M_n$ be a random…

For a $\{-1,1\}$ matrix $M$, let $\operatorname{Res}(M)$ be the minimum number of entries that must be switched from $1$ to $-1$ or vice versa in order to make $M$ singular. Let…

Let $K_n$ be the complete graph on $n$ vertices and let $P_t$ be the path on $t$ vertices. Their direct product $K_n\times P_t$ has vertex set $V(K_n)\times V(P_t)$, with $(g,h)$ a…

Let $M\in\{\pm1\}^{n\times m}$ with $m\geq n$, and let $\operatorname{Res}(M)$ be the least number of entry-flips needed to produce from $M$ a matrix whose rank is strictly less th…

Let $T$ be a first-order theory. Recall that $T$ is resilient if, whenever $(a_i)_{i\in\mathbb{Z}}$ is an indiscernible sequence and $\varphi(x,a_0)$ divides over $a_{\neq 0}$, the…

Global resilience conjecture. Then $G$ has global resilience $(1/4+o(1))nd$ with respect to being $C_k$-free.