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 the number of transversals in latin squares

The logarithm of the maximum number of transversals over all latin squares of order $n$ is greater than $\frac{n}{6}(\ln n+ O(1))$

Upper bounds on the numbers of binary plateaued and bent functions

The logarithm of the number of binary n-variable bent functions is asymptotically less than $(2^n)/3$ as n tends to infinity. Keywords: boolean function, Walsh--Hadamard transform, plateaued function, bent function, upper boun

A Lower Bound on the Number of Boolean Functions with Median Correlation Immunity

The number of $n$-ary balanced correlation immune (resilient) Boolean functions of order $\frac{n}{2}$ is not less than $n^{2^{(n/2)-2}(1+o(1))}$ as $n\rightarrow\infty$. Keywords: resilient function, correlation immune function, orthogonal arrayComment: 3 page