A coloring of the vertices of a graph is called perfect if the multiset of colors of all neighbors of a vertex depends only on its own color. We study the possible parameters of perfect 2-colorings of the n-dimensional cube. Some necessary conditions are obtained for existence of such colorings. A new recursive construction of such colorings is found, which produces colorings for all known and infinitely many new parameter sets.The notion of a perfect coloring of a graph arises naturally in graph theory, algebraic combinatorics (distance regular graphs), and coding theory (perfect codes). A coloring of vertices of a graph is called perfect if, for every vertex, the collection of colors of its neighbors depends only on its own color. Other terms used for this notion in the literature are "equitable partition," "partition design," and "distributive coloring." In [1] we can find a good introduction to this notion in its relation to distance regular graphs. One of the classical notions in coding theory, a perfect code, is a special case of a perfect coloring of a hypercube. We can also mention the paper [2] in which perfect 2-colorings of the infinite square grid are classified; and [3], describing the possible parameters of perfect 2-colorings of plane triangulations.In the present article we address the question of existence of perfect 2-colorings of hypercubes with given parameters. We find a general construction of such colorings (covering, in particular, all parameter sets for which such colorings are known) and give several necessary conditions. The question of classifying such colorings up to isomorphism is outside our scope.

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Let $V$ and $W$ be $n$-dimensional vector spaces over $GF(2)$. A mapping $Q:V\rightarrow W$ is called crooked if it satisfies the following three properties: $Q(0)=0$; $Q(x)+Q(y)+Q(z)+Q(x+y+z)\neq 0$ for any three distinct $x,y,z$; $Q(x)+Q(y)+Q(z)+Q(x+a)+Q(y+a)+Q(z+a)\neq 0$ if $a\neq 0$ ($x,y,z$ arbitrary). We show that every crooked function gives rise to a distance regular graph of diameter 3 having $\lambda=0$ and $\mu=2$ which is a cover of the complete graph. Our approach is a generalization of a recent construction found by de Caen, Mathon, and Moorhouse. We study graph-theoretical properties of the resulting graphs, including their automorphisms. Also we demonstrate a connection between crooked functions and bent functions.

Consider natural numbers $\{1, \cdots, n\}$ colored in three colors. We prove that if each color appears on at least $(n+4)/6$ numbers then there is a three-term arithmetic progression whose elements are colored in distinct colors. This variation on the theme of Van der Waerden's theorem proves the conjecture of Jungić et al.

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