14 problems

Let $k$ and $t$ be fixed, and let $D=\binom{k}{t}-1$. Consider a covering produced by the random greedy algorithm, and define its density as the number of blocks divided by the asy…

Erdős–Lovász linearity conjecture. The true answer is linear, so that

Let $S elangle0,1|^ranglen$ be a symmetric subset of the Boolean hypercube, and let $Wn(S)$ be its set of Hamming weights. Let $W{n,2}$ denote the corresponding weight set for the…

Explicit asymptotic formula conjecture. If $d$ is odd and

Linear-growth conjecture. For all fixed $d$ and $k$, there exists a constant $C_{d,k}$ such that

Three-dimensional covering conjecture. For all $n,k\geq 1$,

Two-dimensional asymptotic conjecture. For all positive integers $n$ and $k$,

Fractional lower-bound conjecture. For every fixed $k$, the bound $kf^*(n,2)$ is not best possible for all sufficiently large $n$; equivalently, the integer covering number satisfi…

Falgas-Ravry–Zhao's conjecture. The gap between the known lower and upper bounds for $c_2(n,K_4^-)$ can be closed; equivalently, the exact minimum codegree threshold for $K_4^-$-co…

Let $n$ be an odd positive integer, and let $\xi(n)$ denote the toroidal covering parameter. Multiplicative reduction conjecture. … This conjecture proposes an exact reduction of t…

Let $\xi(n)$ denote the toroidal covering parameter studied in the paper, and let $p$ be a prime number. Prime-index toroidal covering conjecture. … The paper presents this as one…

Let $G$ be an $n$-vertex $k$-uniform hypergraph, and let $\chi^c(G,k,r,3)$ denote the minimum number of colorings needed for a strong $(r,3)$ cover of $G$. Color-increase conjectur…

Let $\mathrm{C}(n,k)$ denote the minimum number of $k$-element blocks in a covering system on an $n$-element ground set, and let $\mathrm{C}(n,n-4)$ be the covering number in the p…

Let $S_n$ be the set of permutations of $\{1,\ldots,n\}$, and let $\kappa_n$ denote the minimum number of $(n+1)$-permutations needed to cover every $n$-permutation. Asymptotic cov…