# Alantha Newman and Alexandar Nikolov Disprove Beck’s 3-Permutations Conjecture

Alantha Newman and Alexandar Nikolov disproved a few months ago one of the most famous and frustrating open problem in discrepancy theory: Beck’s 3-permutations conjecture. Their paper  A counterexample to Beck’s conjecture on the discrepancy of three permutations is already on the arxive since April. (I was slow to get the news. Yuval Peres who heard it from Prasad Tetali told me about it today. You can read about it already on Joel Spencer’s homepage. Joel had offered 100\$ prize for solving the conjecture that Alantha and Alexandar collected.)

The paper’s abstract tells the story.

Our previos post gives the basic definition of discrepancy of a hypergraph (=set-system), and describes a theorem by Beck and Fiala.

Abstract: Given three permutations on the integers 1 through n, consider the set system consisting of each interval in each of the three permutations. Jozsef Beck conjectured (c. 1987) that the discrepancy of this set system is O(1). We give a counterexample to this conjecture: for any positive integer $n = 3^k$, we exhibit three permutations whose corresponding set system has discrepancy $\Omega (\log (n))$. Our counterexample is based on a simple recursive construction, and our proof of the discrepancy lower bound is by induction. This example also disproves a generalization of Beck’s conjecture due to Spencer, Srinivasan and Tetali, who conjectured that a set system corresponding to l permutations has discrepancy $O(\sqrt l)$.

# Discrepancy, The Beck-Fiala Theorem, and the Answer to “Test Your Intuition (14)”

## The Question

Suppose that you want to send a message so that it will reach all vertices of the discrete $n$-dimensional cube. At each time unit (or round) you can send the message to one vertex. When a vertex gets the message at round $i$ all its neighbors will receive it at round $i+1$.

We will denote the vertices of the discrete $n$-cube by $\pm 1$ vectors of length $n$. The Hamming distance $d(x,y)$ between two such vertices is the number of coordinates they differ and two vertices are neighbors if their Hamming distance equals 1.

The question is

how many rounds it will take to transmit the message to all vertices?

Here is a very simple way to go about it: At round 1 you send the message to vertex (-1,-1,-1,…,-1). At round 2 you send the message to vertex (1,1,…,1). Then you sit and wait. With this strategy it will take $\lceil n/2\rceil+1$ rounds.

Test your intuition: Is there a better strategy?

The answer to the question posed in test you intuition (14) is no. There is no better strategy. The question was originated in Intel. It was posed by Joe Brandenburg and David Scott, and popularized by Vance Faber. The answer was proved by Noga Alon in the paper Transmitting in the n-dimensional cube, Discrete Applied Math. 37/38 (1992), 9-11. The result is closely related to combinatorial discrepancy theory and the proof is related to the argument in the Beck-Fiala theorem and a related lemma by Beck and Spencer. This is a good opportunity to present the Beck-Fiala Theorem.

## The general form of a discrepancy problem

Let $H$ be a hypergraph, i.e., a collection of subsets of a ground set $A$. $A$ is also called the set of vertices of the hypergraphs and the subsets in $H$ are also called edges.

The discrepancy of $H$, denoted by $disc(H)$  is the minimum over all functions $f:A \to \{-1,1\}$ of the maximum over all  $S \in H$ of

$\sum \{f(s):s\in S\}$.

The Erdős Discrepancy Problem (EDP) asks about the following particular hypergraph: The vertices are {1,2,…,n} and the edges are sets of vertices which form arithmetic progressions of the form (r,2r,3r,…kr}. EDP was extensively discussed and studied in polymath5. Here is the link of the first post. Here are links to all polymath5 posts.  Here is the link to polymath5 wiki.

## The Beck-Fiala theorem

Theorem (Beck-Fiala): If every element in $A$ is included in at most $t$ edges  of  $H$ then $disc(H)<2t$.

Before proving this theorem we mention that it is a famous open conjecture to show that actually $disc(H)=O(\sqrt t)$

# Test Your Intuition (14): A Discrete Transmission Problem

Recall that the $n$-dimensional discrete cube is the set of all binary vectors ($0-1$ vectors) of length n. We say that two binary vectors are adjacent if they differ in precisely one coordinate. (In other words, their Hamming distance is 1.) This gives the $n$-dimensional discrete cube a structure of a graph, so from now on , we will refer to binary vectors as vertices.

Suppose that you want to send a message so that it will reach all vertices of the discrete $n$-dimensional cube. At each time unit (or round) you can send the message to one vertex. When a vertex gets the message at round $i$ all its neighbors will receive it at round $i+1$.

The question is

how many rounds it will take to transmit the message to all vertices?

Here is a very simple way to go about it: At round 1 you send the message to vertex (0,0,0,…,0). At round 2 you send the message to vertex (1,1,…,1). Then you wait and sit. With this strategy it will take $\lceil n/2\rceil+1$ rounds.

Test your intuition: Is there a better strategy?