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A Discrete Variation of Littlewood--Offord Problem

Hossein Esmailian, Ebrahim Ghorbani

Abstract

Littlewood--Offord Problem concerns the number of subsums of a set of vectors that fall in a given convex set. We present a discrete variation of this problem where we estimate the number of subsums that are $(0,1)$-vectors. We then utilize this to find the maximum order of graphs with given rank or corank. The rank of a graph $G$ is the rank of its adjacency matrix $A(G)$ and the corank of $G$ is the rank of $A(G)+I$.

A Discrete Variation of Littlewood--Offord Problem

Abstract

Littlewood--Offord Problem concerns the number of subsums of a set of vectors that fall in a given convex set. We present a discrete variation of this problem where we estimate the number of subsums that are -vectors. We then utilize this to find the maximum order of graphs with given rank or corank. The rank of a graph is the rank of its adjacency matrix and the corank of is the rank of .

Paper Structure

This paper contains 7 sections, 12 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Littlewood--Offord Problem and its variants
  3. Rank-order problems for graphs
  4. Discrete Variation of Littlewood--Offord Problem
  5. Applications
  6. Bipartite graphs
  7. Cobipartite graphs

Key Result

Theorem 1

Let $x_1,\ldots,x_\ell$ be real numbers with $|x_i|\ge1$ for all $i$. Let $\Lambda$ be an open interval of length $1$. Then the total number of $\ell$-tuples $(\epsilon_1, \ldots,\epsilon_\ell) \in \{0, 1\}^\ell$ with $\epsilon_1x_1 + \cdots+\epsilon_\ell x_\ell\in \Lambda$ is at most ${\ell\choose\

Figures (1)

  • Figure 1: The graph $\mathcal{D}_3$.

Theorems & Definitions (20)

  • Theorem 1: Erdős erd
  • Theorem 2: Kleitman kl70
  • Theorem 3
  • Conjecture 4: Akbari, Cameron and Khosrovshahi ack
  • Remark 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 10 more