10 problems

Let $P$ be a permutation matrix, and let $\operatorname{ex}(n,P)$ denote the extremal number of $P$. Füredi–Hajnal conjecture. For every permutation matrix $P$, … This conjecture i…

Let $t\geq 2$, and let $A$ be a zero-one matrix. The matrix is assumed to contain at most $t$ $1$-entries in each row. Let $\operatorname{ex}(n,A)$ denote its extremal number. Meth…

Let … be the three-dimensional $0$-$1$ matrix described above, and let $\operatorname{ex}(n,P)$ denote the maximum number of $1$-entries in an $n\times n\times n$ $0$-$1$ matrix av…

Let $M(n,k)$ be the maximum of $\lvert\det(A)\rvert$ over all $n\times n$ zero-one matrices whose rows each sum to $k$. For $k=2$, Bruhn and Rautenbach proved the exponential bound…

Let $A$ be a zero-one matrix corresponding to a cycle of length $6$. Six-cycle exponent conjecture. … For every such matrix, the paper gives a lower bound $\Omega(n^{4/3})$ and an…

Let $t\geq 2$ be a positive integer, and let $A$ be a zero-one matrix such that every row of $A$ contains at most $t$ $1$-entries. Maximum-row-degree exponent conjecture. … This is…

Let $t\geq 2$ be an integer, and let $A$ be a column-$t$-partite zero-one matrix. Sharp exponent conjecture. … The paper proves this conjecture when $A$ is $t\times t$-partite. It…

Let $T(n,\tilde{k})$ be the family of matrices with real row-weight parameter $\tilde{k}>1$, and let $M_T(n,\tilde{k})$ be its maximum determinant. Define … and … Let $M_R(m,n,k)$…

Bruhn et al.'s conjecture. The true exponential growth rate is

Let $S(n,k)$ denote the family of $n\times n$ zero-one matrices with every row and column sum equal to $k$, and let $M(n,k)$ be the maximum absolute determinant of a matrix in…