---
title: "Discrepancy of hypergraphs"
chunk: 1/2
source: "https://en.wikipedia.org/wiki/Discrepancy_of_hypergraphs"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T11:02:54.676918+00:00"
instance: "kb-cron"
---

Discrepancy of hypergraphs is an area of discrepancy theory that studies the discrepancy of general set systems.

== Definitions ==
In the classical setting, we aim at partitioning the vertices of a hypergraph 
  
    
      
        
          
            H
          
        
        =
        (
        V
        ,
        
          
            E
          
        
        )
      
    
    {\displaystyle {\mathcal {H}}=(V,{\mathcal {E}})}
  
 into two classes in such a way that ideally each hyperedge contains the same number of vertices in both classes. A partition into two classes can be represented by a coloring 
  
    
      
        χ
        :
        V
        →
        {
        −
        1
        ,
        +
        1
        }
      
    
    {\displaystyle \chi \colon V\rightarrow \{-1,+1\}}
  
. We call −1 and +1 colors. The color-classes 
  
    
      
        
          χ
          
            −
            1
          
        
        (
        −
        1
        )
      
    
    {\displaystyle \chi ^{-1}(-1)}
  
 and 
  
    
      
        
          χ
          
            −
            1
          
        
        (
        +
        1
        )
      
    
    {\displaystyle \chi ^{-1}(+1)}
  
 form the corresponding partition. For a hyperedge 
  
    
      
        E
        ∈
        
          
            E
          
        
      
    
    {\displaystyle E\in {\mathcal {E}}}
  
, set

  
    
      
        χ
        (
        E
        )
        :=
        
          ∑
          
            v
            ∈
            E
          
        
        χ
        (
        v
        )
        .
      
    
    {\displaystyle \chi (E):=\sum _{v\in E}\chi (v).}
  

The discrepancy of 
  
    
      
        
          
            H
          
        
      
    
    {\displaystyle {\mathcal {H}}}
  
 with respect to 
  
    
      
        χ
      
    
    {\displaystyle \chi }
  
 and the discrepancy of 
  
    
      
        
          
            H
          
        
      
    
    {\displaystyle {\mathcal {H}}}
  
 are defined by

  
    
      
        disc
        ⁡
        (
        
          
            H
          
        
        ,
        χ
        )
        :=
        
        
          max
          
            E
            ∈
            
              
                E
              
            
          
        
        
          |
        
        χ
        (
        E
        )
        
          |
        
        ,
      
    
    {\displaystyle \operatorname {disc} ({\mathcal {H}},\chi ):=\;\max _{E\in {\mathcal {E}}}|\chi (E)|,}
  

  
    
      
        disc
        ⁡
        (
        
          
            H
          
        
        )
        :=
        
          min
          
            χ
            :
            V
            →
            {
            −
            1
            ,
            +
            1
            }
          
        
        disc
        ⁡
        (
        
          
            H
          
        
        ,
        χ
        )
        .
      
    
    {\displaystyle \operatorname {disc} ({\mathcal {H}}):=\min _{\chi :V\rightarrow \{-1,+1\}}\operatorname {disc} ({\mathcal {H}},\chi ).}
  

These notions as well as the term 'discrepancy' seem to have appeared for the first time in a paper of Beck. Earlier results on this problem include the famous lower bound on the discrepancy of arithmetic progressions by Roth and upper bounds for this problem and other results by Erdős and Spencer and Sárközi. At that time, discrepancy problems were called quasi-Ramsey problems.

== Examples ==
To get some intuition for this concept, let's have a look at a few examples.

If all edges of 
  
    
      
        
          
            H
          
        
      
    
    {\displaystyle {\mathcal {H}}}
  
 intersect trivially, i.e. 
  
    
      
        
          E
          
            1
          
        
        ∩
        
          E
          
            2
          
        
        =
        ∅
      
    
    {\displaystyle E_{1}\cap E_{2}=\varnothing }
  
 for any two distinct edges 
  
    
      
        
          E
          
            1
          
        
        ,
        
          E
          
            2
          
        
        ∈
        
          
            E
          
        
      
    
    {\displaystyle E_{1},E_{2}\in {\mathcal {E}}}
  
, then the discrepancy is zero, if all edges have even cardinality, and one, if there is an odd cardinality edge.
The other extreme is marked by the complete hypergraph 
  
    
      
        (
        V
        ,
        
          2
          
            V
          
        
        )
      
    
    {\displaystyle (V,2^{V})}
  
. In this case the discrepancy is 
  
    
      
        ⌈
        
          
            1
            2
          
        
        
          |
        
        V
        
          |
        
        ⌉
      
    
    {\displaystyle \lceil {\frac {1}{2}}|V|\rceil }
  
. Any 2-coloring will have a color class of at least this size, and this set is also an edge. On the other hand, any coloring 
  
    
      
        χ
      
    
    {\displaystyle \chi }
  
 with color classes of size 
  
    
      
        ⌈
        
          
            1
            2
          
        
        
          |
        
        V
        
          |
        
        ⌉
      
    
    {\displaystyle \lceil {\frac {1}{2}}|V|\rceil }
  
 and 
  
    
      
        ⌊
        
          
            1
            2
          
        
        
          |
        
        V
        
          |
        
        ⌋
      
    
    {\displaystyle \lfloor {\frac {1}{2}}|V|\rfloor }
  
 proves that the discrepancy is not larger than 
  
    
      
        ⌈
        
          
            1
            2
          
        
        
          |
        
        V
        
          |
        
        ⌉
      
    
    {\displaystyle \lceil {\frac {1}{2}}|V|\rceil }
  
. It seems that the discrepancy reflects how chaotic the hyperedges of 
  
    
      
        
          
            H
          
        
      
    
    {\displaystyle {\mathcal {H}}}
  
 intersect. Things are not that easy, however, as the following example shows.
Set 
  
    
      
        n
        =
        4
        k
      
    
    {\displaystyle n=4k}
  
, 
  
    
      
        k
        ∈
        
          
            N
          
        
      
    
    {\displaystyle k\in {\mathcal {N}}}
  
 and 
  
    
      
        
          
            
              H
            
          
          
            n
          
        
        =
        (
        [
        n
        ]
        ,
        {
        E
        ⊆
        [
        n
        ]
        ∣
        
          |
        
        E
        ∩
        [
        2
        k
        ]
        
          |
        
        =
        
          |
        
        E
        ∖
        [
        2
        k
        ]
        
          |
        
        }
        )
      
    
    {\displaystyle {\mathcal {H}}_{n}=([n],\{E\subseteq [n]\mid |E\cap [2k]|=|E\setminus [2k]|\})}
  
. In words, 
  
    
      
        
          
            
              H
            
          
          
            n
          
        
      
    
    {\displaystyle {\mathcal {H}}_{n}}
  
 is the hypergraph on 4k vertices {1,...,4k}, whose edges are all subsets that have the same number of elements in {1,...,2k} as in {2k+1,...,4k}. Now 
  
    
      
        
          
            
              H
            
          
          
            n
          
        
      
    
    {\displaystyle {\mathcal {H}}_{n}}
  
 has many (more than 
  
    
      
        
          
            
              
                (
              
              
                
                  n
                  
                    /
                  
                  2
                
                
                  n
                  
                    /
                  
                  4
                
              
              
                )
              
            
          
          
            2
          
        
        =
        Θ
        (
        
          
            1
            n
          
        
        
          2
          
            n
          
        
        )
      
    
    {\displaystyle {\binom {n/2}{n/4}}^{2}=\Theta ({\frac {1}{n}}2^{n})}
  
) complicatedly intersecting edges. However, its discrepancy is zero, since we can color {1,...,2k} in one color and {2k+1,...,4k} in another color.
The last example shows that we cannot expect to determine the discrepancy by looking at a single parameter like the number of hyperedges. Still, the size of the hypergraph yields first upper bounds.

== General hypergraphs ==
1. For any hypergraph  
  
    
      
        
          
            H
          
        
      
    
    {\displaystyle {\mathcal {H}}}
  
  with n vertices and m edges:

  
    
      
        disc
        ⁡
        (
        
          
            H
          
        
        )
        ≤
        
          
            2
            n
            ln
            ⁡
            (
            2
            m
            )
          
        
        .
      
    
    {\displaystyle \operatorname {disc} ({\mathcal {H}})\leq {\sqrt {2n\ln(2m)}}.}
  

The proof is a simple application of the probabilistic method. Let 
  
    
      
        χ
        :
        V
        →
        {
        −
        1
        ,
        1
        }
      
    
    {\displaystyle \chi :V\rightarrow \{-1,1\}}
  
 be a random coloring, i.e. we have

  
    
      
        Pr
        (
        χ
        (
        v
        )
        =
        −
        1
        )
        =
        Pr
        (
        χ
        (
        v
        )
        =
        1
        )
        =
        
          
            1
            2
          
        
      
    
    {\displaystyle \Pr(\chi (v)=-1)=\Pr(\chi (v)=1)={\frac {1}{2}}}
  

independently for all 
  
    
      
        v
        ∈
        V
      
    
    {\displaystyle v\in V}
  
. Since 
  
    
      
        χ
        (
        E
        )
        =
        
          ∑
          
            v
            ∈
            E
          
        
        χ
        (
        v
        )
      
    
    {\displaystyle \chi (E)=\sum _{v\in E}\chi (v)}
  
 is a sum of independent −1, 1 random variables. So we have 
  
    
      
        Pr
        (
        
          |
        
        χ
        (
        E
        )
        
          |
        
        >
        λ
        )
        <
        2
        exp
        ⁡
        (
        −
        
          λ
          
            2
          
        
        
          /
        
        (
        2
        n
        )
        )
      
    
    {\displaystyle \Pr(|\chi (E)|>\lambda )<2\exp(-\lambda ^{2}/(2n))}
  
 for all 
  
    
      
        E
        ⊆
        V
      
    
    {\displaystyle E\subseteq V}
  
 and 
  
    
      
        λ
        ≥
        0
      
    
    {\displaystyle \lambda \geq 0}
  
. Taking 
  
    
      
        λ
        =
        
          
            2
            n
            ln
            ⁡
            (
            2
            m
            )
          
        
      
    
    {\displaystyle \lambda ={\sqrt {2n\ln(2m)}}}
  
 gives

  
    
      
        Pr
        (
        disc
        ⁡
        (
        
          
            H
          
        
        ,
        χ
        )
        >
        λ
        )
        ≤
        
          ∑
          
            E
            ∈
            
              
                E
              
            
          
        
        Pr
        (
        
          |
        
        χ
        (
        E
        )
        
          |
        
        >
        λ
        )
        <
        1.
      
    
    {\displaystyle \Pr(\operatorname {disc} ({\mathcal {H}},\chi )>\lambda )\leq \sum _{E\in {\mathcal {E}}}\Pr(|\chi (E)|>\lambda )<1.}