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We construct a symmetric invertible binary pairing function $F(m,n)$ on the set of positive integers with a property of $F(m,n)=F(n,m)$. Then we provide a complete proof of its symmetry and bijectivity, from which the construction of symmetric invertible binary pairing functions on any custom set of integers could be seen.
A ${00,01,10,11}$-valued function on the vertices of the $n$-cube is called a $t$-resilient $(n,2)$-function if it has the same number of $00$s, $01$s, $10$s and $11$s among the vertices of every subcube of dimension $t$. The Friedman and Fon-Der-Fla
We show that a Boolean degree $d$ function on the slice $binom{[n]}{k} = { (x_1,ldots,x_n) in {0,1} : sum_{i=1}^n x_i = k }$ is a junta, assuming that $k,n-k$ are large enough. This generalizes a classical result of Nisan and Szegedy on the hypercube
We present results on the existence of long arithmetic progressions in the Thue-Morse word and in a class of generalised Thue-Morse words. Our arguments are inspired by van der Waerdens proof for the existence of arbitrary long monochromatic arithmet
A fringe subtree of a rooted tree is a subtree consisting of one of the nodes and all its descendants. In this paper, we are specifically interested in the number of non-isomorphic trees that appear in the collection of all fringe subtrees of a binar
We show that if $fcolon S_n to {0,1}$ is $epsilon$-close to linear in $L_2$ and $mathbb{E}[f] leq 1/2$ then $f$ is $O(epsilon)$-close to a union of mostly disjoint cosets, and moreover this is sharp: any such union is close to linear. This constitute