DP-colorings of uniform hypergraphs and splittings of Boolean hypercube into faces

We develop a connection between DP-colorings of $k$-uniform hypergraphs of order $n$ and coverings of $n$-dimensional Boolean hypercube by pairs of antipodal $(n-k)$-dimensional faces. Bernshteyn and Kostochka established that the lower bound on edges in a non-2-DP-colorable $k$-uniform hypergraph is equal to $2^{k-1}$ for odd $k$ and $2^{k-1}+1$ for even $k$. They proved that these bounds are tight for $k=3,4$. In this paper, we prove that the bound is achieved for all odd $k\geq 3$.Comment: The previous versions of paper contains a significant erro

On unbalanced Boolean functions with best correlation immunity

It is known that the order of correlation immunity of a nonconstant unbalanced Boolean function in $n$ variables cannot exceed $2n/3-1$; moreover, it is $2n/3-1$ if and only if the function corresponds to an equitable $2$-partition of the $n$-cube with an eigenvalue $-n/3$ of the quotient matrix. The known series of such functions have proportion $1:3$, $3:5$, or $7:9$ of the number of ones and zeros. We prove that if a nonconstant unbalanced Boolean function attains the correlation-immunity bound and has ratio $C:B$ of the number of ones and zeros, then $CB$ is divisible by $3$. In particular, this proves the nonexistence of equitable partitions for an infinite series of putative quotient matrices. We also establish that there are exactly $2$ equivalence classes of the equitable partitions of the $12$-cube with quotient matrix $[[3,9],[7,5]]$ and $16$ classes, with $[[0,12],[4,8]]$. These parameters correspond to the Boolean functions in $12$ variables with correlation immunity $7$ and proportion $7:9$ and $1:3$, respectively (the case $3:5$ remains unsolved). This also implies the characterization of the orthogonal arrays OA$(1024,12,2,7)$ and OA$(512,11,2,6)$.Comment: v3: final; title changed; revised; OA(512,11,2,6) discusse