Improved Bounds for Uniform Hypergraphs without Property B

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by  Sachin Aglave

Improved Bounds for Uniform Hypergraphs without Property B
Sachin Aglave∗ V. A. Amarnath† Saswata Shannigrahi‡§ Shwetank Singh¶
January 9, 2017
7
1
0
2
Abstract
n
a A hypergraph is said to be properly 2-colorable if there exists a 2-coloring of its vertices
J
such that no hyperedge is monochromatic. On the other hand, a hypergraph is called non-2-

6 colorableifthereexistsatleastonemonochromatichyperedgeineachofthepossible2-colorings

of its vertex set. Let m(n) denote the minimum number of hyperedges in a non-2-colorable n-
]
O uniformhypergraph. Establishingthelowerandupperboundsonm(n)isawell-studiedresearch
C direction over several decades. In this paper, we improve the upper bounds for m(8), m(13),
m(14),m(16)andm(17)byconstructingnon-2-colorableuniformhypergraphs. Wealsoimprove
.
h
the lower bound for m(5).
t
a
m Keywords: Property B; Uniform Hypergraphs; Hypergraph 2-coloring
[

2 1 Introduction
v
8
Hypergraphs are combinatorial structures that are generalizations of graphs. Let H = (V,E) be an
1
2 n-uniform hypergraph with vertex set V, with each hyperedge in E having exactly n vertices in it.
0 A 2-coloring of H is an assignment of one of the two colors red and blue to each of the vertices in
0
V. We say a 2-coloring of H to be proper if each of its hyperedges has red as well as blue vertices.
.
2
H is said to be non-2-colorable if no proper 2-coloring exists for it; otherwise, it is said to satisfy
0
6 Property B. For an integer n 1, let m(n) denote the minimum number of hyperedges present in
≥
1 a non-2-colorable n-uniform hypergraph.
:
v Establishing an upper bound on m(n) is a well-explored combinatorial problem. Erd˝os [6] gave
Xi a non-constructive proof to establish the currently best-known upper bound m(n) = O(n22n).
However, there is no known construction for a non-2-colorable n-uniform hypergraph that matches
r
a
this upper bound. Abbott and Moser [2] constructed a non-2-colorable n-uniform hypergraph with
O((√7 + o(1))n) hyperedges. Recently, Gebauer [8] improved this result by constructing a non-
2-colorable n-uniform hypergraph with O(2(1+o(1))n) hyperedges. Even though this is the best
construction known for a non-2-colorable n-uniformhypergraphfor large n, it is still asymptotically
far from the above-mentioned non-constructive upper bound given by Erd˝os.
Finding upper bounds for small values of n is also a well-studied problem and several construc-
tions have been given for establishing these. For example, it can be easily seen that m(1) 1,
≤
m(2) 3 (triangle graph) and m(3) 7 (Fano plane). Moreover, the previously mentioned con-
≤ ≤
struction of Abbott and Moser shows that m(6) 147. For n 3, Abbott and Hanson [1]
≤ ≥
∗IITGuwahati, India. Email: [email protected]
†IITGuwahati, India. Email: [email protected]
‡IIITHyderabad,India. Email: [email protected]
§Corresponding Author
¶IITGuwahati, India. Email: [email protected]
1
n m(n) Corresponding construction/recurrence relation
1 m(1) = 1 Single Vertex
2 m(2) = 3 Triangle Graph
3 m(3) = 7 Fano Plane [10]
4 m(4) = 23 [12], [14]
5 m(5) 51 m(5) 24 +5m(3)
≤ ≤
6 m(6) 147 m(6) m(2)m(3)2
≤ ≤
7 m(7) 421 m(7) 26 +7m(5)
≤ ≤
8 m(8) 1269 m(8) 2m(3)m(5)+8m(3)2 +27+ 8 /2
≤ ≤ 4
9 m(9) 2401 m(9) m(3)4
≤ ≤ (cid:0) (cid:1)
10 m(10) 7803 m(10) m(2)m(5)2
≤ ≤
11 m(11) 25449 m(11) 15 28 +9m(9)
≤ ≤ ·
12 m(12) 55223 m(12) m(3)4m(4)2
≤ ≤
13 m(13) 297347 m(13) 17 210 +11m(11)
≤ ≤ ·
14 m(14) 531723 m(14) m(2)m(7)2
≤ ≤
15 m(15) 857157 m(15) m(3)5m(5)
≤ ≤
16 m(16) 4831083 m(16) m(2)m(8)2
≤ ≤
17 m(17) 13201419 m(17) 21 214 +15m(15)
≤ ≤ ·
Table 1: Best-known upper bounds on m(n) for small values of n
gave a construction using a non-2-colorable (n 2)-uniform hypergraph to show that m(n)
− ≤
n m(n 2)+2n−1+2n−2((n 1) mod 2). Using the best-known upper bounds on m(n 2), this
· − − −
recurrencegives non-trivialupperboundsonm(n)forafewsmallvaluesofn. Forexample, itshows
that m(4) 24, m(5) 51 and m(7) 421. Seymour [14] improved the upper bound on m(4)
≤ ≤ ≤
to m(4) 23 by constructing a non-2-colorable 4-uniform hypergraph with 23 hyperedges. For
≤
even integers n 4, Toft [15] generalized this construction using a non-2-colorable (n 2)-uniform
≥ −
hypergraph to improve Abbott and Hanson’s result to m(n) n m(n 2)+2n−1 + n /2. In
≤ · − n/2
particular, this led to establishing an upper bound m(8) 1339. For a given integer n 3 and a
≤ (cid:0)≥ (cid:1)
non-2-colorable (n 2)-uniform hypergraph A, we refer to Abbott-Hanson’s construction for odd
−
n and Toft’s construction for even n as Abbott-Hanson-Toft construction and denote the number
of hyperedges in such a hypergraph as m (n). It can be easily observed that m(n) m (n) for
A A
≤
any non-2-colorable (n 2)-uniform hypergraph A. In fact, we have already seen that the above-
−
mentioned upper bounds m(4) 23, m(5) 51, m(7) 421 and m(8) 1339 are obtained by
≤ ≤ ≤ ≤
Abbott-Hanson-Toft constructions using the best-known constructions for non-2-colorable 2, 3, 5
and 6-uniform hypergraphs, respectively. Recently, the construction given by Mathews et al. [11]
improvedtheupperboundonm(8)tom(8) 1269. Inaddition, theymodifiedtheAbbott-Hanson-
≤
Toft construction to improve the upper bounds on m(n) for some small values of n in the range
9 n 17. The currently best-known upper bounds on m(n) for n 17 are given in Table 1.
≤ ≤ ≤
In the other direction, Erd˝os [6] showed the lower bound on m(n) to be m(n) = Ω(2n), which
was later improved by Beck [3] to m(n) = Ω(n1/3−o(1)2n). The currently best-known lower bound
m(n) = Ω( n 2n) was given by Radhakrishnan and Srinivasan [13]. Recently, a simpler proof
lnn
for the same result has been given by Cherkashin and Kozik [4]. Note that there is a significant
p
asymptotic gap between thecurrently best-known lower and upperboundson m(n). Even for small
values of n,weare only aware of afewlower boundsfor m(n)that matches thecorrespondingupper
bounds. It can be easily seen that m(1) 1, m(2) 3 and m(3) 7 and therefore m(1) = 1,
≥ ≥ ≥
2
m(2) = 3andm(3) = 7. Recently, O¨sterg˚ard[12]showedthatm(4) 23andestablishedm(4) = 23
≥
as a result. The exact values of m(n) are not yet known for n 5.
≥
1.1 Our Contributions
In this paper, we give constructions that improve the best-known upper bounds on m(8), m(13),
m(14), m(16) and m(17). We also establish a non-trivial lower bound on m(5).
In Section 3.1, we give a construction that gives the following recurrence relation. In particular,
it improves the upper bound on m(13).
Result 1. Consider an integer k 1. For an odd n > 2k, m(n) n+k−1 m(n 2k)+ n+k−1 2n−1.
≥ ≤ k − k−1
For an even n > 2k, m(n) n+k−1 m(n 2k)+ n+k−1 (2n−1+ n /2).
≤ k − k−1 (cid:0)n/2 (cid:1) (cid:0) (cid:1)
This construction also give(cid:0)s a non(cid:1)-2-colorable n(cid:0)-unifor(cid:1)m hyperg(cid:0)rap(cid:1)h with O(3.76n) hyperedges.
Even though we note that it gives a better constructive upper bound m(n) = O(3.76n) than the
trivialboundm(n) 2n−1 = Θ(4n/√n), itisasymptotically worsethan thepreviously mentioned
≤ n
constructive upper bounds m(n)= O((√7+o(1))n) [2] and m(n) = O(2(1+o(1))n) [8].
(cid:0) (cid:1)
InSection 3.2,weprovideanotherconstructionthatimprovestheupperboundsonm(8), m(13),
m(14), m(16) and m(17).
Result 2. Consider an integer k satisfying 0 < k < n. Let w = n/k , x = n mod k, y = k/x
⌊ ⌋ ⌊ ⌋
and z = k mod x.
(a) If x > 0 and z > 0, m(n) w m(n k)m(k)+y m(k)wm(x)+ x+z−1 m(n k)m(x)y +
≤ · − · z −
x+z−1 m(k)w.
x (cid:0) (cid:1)
(b) I(cid:0)f x >(cid:1)0 and z =0, m(n) w m(n k)m(k)+y m(k)wm(x)+m(n k)m(x)y.
≤ · − · −
(c) If x = 0, m(n) w m(n k)m(k)+m(k)w.
≤ · −
In Section 3.3, we give a construction to prove the following result that further improves the
upper bounds on m(13) and m(16).
Result 3. Consider an integer k 2 and a non-2-colorable (k 1)-uniform hypergraph H . Then,
1c
≥ −
m(3k+1) (m(k 1)+2k−1)m(k+1)2 +2m (k+1)m(k)2 +4m(k+1)m(k)2.
≤ − H1c
In Section 4, we improve the currently best-known lower bound m(5) 28.
≥
Result 4. m(5) 29.
≥
2 Previous Results
2.1 Abbott-Moser Construction
AbbottandMoser [2]gave theconstruction for a non-2-colorable n-uniformhypergraphH = (V,E)
by exploiting the known constructions of non-2-colorable a-uniform and b-uniform hypergraphs for
any composite n satisfying n = ab for two integers a 1,b 1.∗ Let H = (V ,E ) and H =
a a a b
≥ ≥
(V ,E ) be non-2-colorable a-uniform and b-uniform hypergraphs, respectively. H is constructed
b b
using V identical copies of H by replacing each vertex of H with a copy of H . Let us denote the
a b a b
| |
∗Note that the notations used in a sub-section are not related to the notations used in other sub-sections, unless
specified otherwise.
3
copies of H as H = (V ,E ),H = (V ,E ),...,H = (V ,E ). The vertex set of H
b b1 b1 b1 b2 b2 b2 b|Va| b|Va| b|Va|
is V = V V V . The hyperedge set of H is constructed as follows. For each hyperedge
b1∪ b2∪···∪ b|Va|
v ,...,v in E , the following collection of hyperedges e e :e E ,...,e E
{ 1 a} a {{ 1∪···∪ a} 1 ∈ bv1 a ∈ bva}
is added to E. The resulting hypergraph H is n-uniform and it is evident from the construction
that it has E E a hyperedges. Abbott and Moser [2] showed that H is non-2-colorable, thereby
a b
| || |
proving the following result.
Lemma 1. For any composite n satisfying n = ab for integers a,b 1, m(n) m(a)m(b)a.
≥ ≤
This construction gives the best-known upper bounds for some small values of n. For example,
it shows that m(6) 147, m(9) 2401, m(10) 7803, m(12) 55223, m(14) 531723, m(15)
≤ ≤ ≤ ≤ ≤ ≤
857157 and m(16) 4831083.
≤
2.2 Abbott-Hanson-Toft Construction
As mentioned in the introduction, Abbott-Hanson’s construction [1] for odd n along with Toft’s
construction [15] for even n is referred to as Abbott-Hanson-Toft construction. For a given n 3,
≥
this construction is built using a non-2-colorable (n 2)-uniform hypergraph, which we call as the
−
core hypergraph and denote by H = (V ,E ). Let its hyperedge set be E = e ,e ,...,e .
c c c c { 1 2 mc}
Let A and B be two disjoint sets of vertices where A= a ,a ,...,a and B = b ,b ,...,b ,
1 2 n 1 2 n
{ } { }
each disjoint with V . For a given K 1,2,...,n , we define A = a , B = b ,
c ⊂ { } K i∈K{ i} K i∈K{ i}
A = A A and B = B B .
K K K K
\ \ S S
The construction of the non-2-colorable n-uniform hypergraph H = (V,E) is as follows. The
vertex set is V = V A B and the hyperedge set E consists of the following hyperedges:
c
∪ ∪
(i) e a b for every pair i,j satisfying 1 i m and 1 j n
i j j c
∪{ }∪{ } ≤ ≤ ≤ ≤
(ii) A B for each K such that K is odd and 1 K n/2
K K
∪ | | ≤ | |≤ ⌊ ⌋
(iii) A B for each K such that K is even and 2 K n/2
K K
∪ | | ≤ | | ≤ ⌊ ⌋
(iv) A
It is easy to observe that the number of hyperedges in H is 2n−1 +nm for odd n and 2n−1 +
c
nm + n /2 for even n. Abbott-Hanson [1] and Toft [15] proved that H is non-2-colorable, and
c n/2
the construction gives the upper bound on m(n) as follows.
(cid:0) (cid:1)
Lemma 2.
2n−1+n m(n 2) if n is odd
m(n) · −
≤ (2n−1+n m(n 2)+ n /2 if n is even
· − n/2
(cid:0) (cid:1)
Lemma 2 gives the best-known upper bounds on m(n) for n = 5 and 7 as m(5) 51 and
≤
m(7) 421, respectively.
≤
2.3 Mathews-Panda-Shannigrahi Construction
Mathews et al. [11] gave three constructions to improve the upper bounds on m(n) for small values
of n.
4
First Construction
The construction described here gives an improvement on the best-known upper bound for m(8).
This construction uses two identical copies of non-2-colorable 3-uniform hypergraphs denoted by
H = (V ,E )andH = (V ,E ),alongwithanon-2-colorable5-uniformhypergraphH = (V ,E ).
1 1 1 2 2 2 3 3 3
Let E = e1,e1,...,e1 , E = e2,e2,...,e2 and E = e3,e3,...,e3 . Consider two disjoint
1 { 1 2 m3} 2 { 1 2 m3} 3 { 1 2 m5}
sets of vertices A and B, each disjoint from V , V and V . Let A = a ,a ,...,a and B =
1 2 3 1 2 8
{ }
b ,b ,...,b . For a given K 1,2,...,8 , we define A = a , B = b , A =
{ 1 2 8} ⊂ { } K i∈K{ i} K i∈K{ i} K
A A and B = B B .
K K K
\ \ S S
Mathews et al. [11] constructed a non-2-colorable 8-uniform hypergraph H = (V,E) as follows.
The vertex set is V = A B V V V and the hyperedge set E consists of the following
1 2 3
∪ ∪ ∪ ∪
hyperedges:
(i) e1 e3 for every pair i,j satisfying 1 i m and 1 j m
i ∪ j ≤ ≤ 3 ≤ ≤ 5
(ii) e2 e3 for every pair i,j satisfying 1 i m and 1 j m
i ∪ j ≤ ≤ 3 ≤ ≤ 5
(iii) A
(iv) A B for each K such that K is odd and 1 K 3
K K
∪ | | ≤ | |≤
(v) A B for each K such that K is even and 2 K 4
K K
∪ | | ≤ | | ≤
(vi) a ,b e1 e2 for every triple i,j,k satisfying 1 i 8, 1 j,k m
{ i i}∪ j ∪ k ≤ ≤ ≤ ≤ 3
Mathews et al. [11] proved that H is non-2-colorable and improved the upper bound on m(8)
to m(8) 1269.
≤
Second Construction
The following construction of a non-2-colorable n-uniform hypergraph for n 3 is an improvement
≥
over theAbbott-Hanson-Toft construction, describedin Section 2.2. Similar to theAbbott-Hanson-
Toft construction, this construction also utilizes a non-2-colorable (n 2)-uniform hypergraphH =
c
−
(V ,E ) that is called the core hypergraph in Section 2.2. Let E = e ,e ,...,e . In addition,
c c c { 1 2 mc}
this construction uses two disjoint vertex sets A = a ,a ,...,a and B = b ,b ,...,b , each
1 2 n 1 2 n
{ } { }
disjoint from V . Let B1 denote the ordered set B1 = (b ,b ,...,b ), where the ordering is defined
c 1 2 n
as b b ... b . For any 1 p n, let Bp denote the ordered set where b and b are
1 2 n 1 p
≺ ≺ ≺ ≤ ≤
swapped in the ordering. Let the ordered set Bp = (b ,b ,...,b ,b ,b ,...,b ) be denoted by
p 2 p−1 1 p+1 n
p p p p p p
(w ,w ,...,w ), where the ordering is given as w w ... w . For K 1,2,...,n , let
1 2 n p p 1 ≺ p2 ≺ ≺p n ⊂ { }
A = a , A = A A , B = w and B = B B .
K i∈K{ i} K \ K K i∈K{ i} K \ K
The construction of the non-2-colorable n-uniform hypergraph H = (V,E) is defined as follows.
S S
The vertex set is V = V A B and the hyperedge set E consists of following hyperedges:
c
∪ ∪
(i) e a b for every pair i,j satisfying 1 i m and 2 j n
i j j c
∪ ∪ ≤ ≤ ≤ ≤
p
(ii) A B foreachpsatisfying1 p n,andeachK suchthat K isoddand1 K n/2
K∪ K ≤ ≤ | | ≤ | |≤ ⌊ ⌋
p
(iii) A B foreachpsatisfying1 p n,andeachK suchthat K isevenand2 K n/2
K∪ K ≤ ≤ | | ≤ | |≤ ⌊ ⌋
(iv) A
It can be seen that the number of hyperedges in H is (n+1)2n−2 +(n 1)m when n is odd
c
−
and (n+1)2n−2 + n /2+(n 1) m + n−2 when n is even. Mathews et al. [11] showed
n/2 − c (n−2)/2
that H is non-2-colorable, which give(cid:16)s the following(cid:17)result.
(cid:0) (cid:1) (cid:0) (cid:1)
5
Lemma 3.
(n+1)2n−2 +(n 1) m(n 2) if n is odd
m(n) − · −
≤ ((n+1)2n−2 + n /2+(n 1) m(n 2)+ n−2 if n is even
n/2 − − (n−2)/2
(cid:16) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1)
This result improved the upper bounds on m(13) and m(17) to m(13) 357892 and m(17)
≤ ≤
14304336, respectively.
Third Construction
Theconstructionpresentedhereisanimprovementoftheconstructiondescribedinprevioussection.
For a given n 3, consider a non-2-colorable (n 2)-uniform hypergraph H = (V ,E ) which is
c c c
≥ −
referred earlier as core hypergraph. Let E = e ,e ,...,e . Let A = a ,a ,...,a , B =
c { 1 2 mc} { 1 2 n}
b ,b ,...,b , A′ = a′,a′,...,a′ and B′ = b′,b′,...,b′ denote disjoint vertex sets, each
{ 1 2 n} { 1 2 n−2} { 1 2 n−2}
disjoint from V . For any K 1,2,...,n , let A = a , B = b , A = A A
c ⊂ { } K i∈K{ i} K i∈K{ i} K \ K
and B = B B . Similarly, for any L 1,2,...,n 2 , let A′ = a′ , B′ = b′ ,
K \ K ⊂ { S− } L Si∈L{ i} L i∈L{ i}
A′ = A′ A′ and B′ = B′ B′ .
L \ L L \ L S S
The construction of the non-2-colorable n-uniform hypergraph H = (V,E) is defined as follows.
Thevertex setV isV A B A′ B′ andthehyperedgesetE consistsof thefollowing hyperedges:
c
∪ ∪ ∪ ∪
(i) e a′ b′ for every pair i,j satisfying 1 i m and 1 j n 2
i ∪{ j}∪{ j} ≤ ≤ c ≤ ≤ −
(ii) A
(iii) A B for each K such that K is odd and 1 K n/2
K K
∪ | | ≤ | |≤ ⌊ ⌋
(iv) A B for each K such that K is even and 2 K n/2
K K
∪ | | ≤ | | ≤ ⌊ ⌋
(v) A′ a b for each i satisfying 1 i n
i i
∪{ }∪{ } ≤ ≤
(vi) A′ B′ a b for each i satisfying 1 i n, and for each L such that L is odd and
L∪ L∪{ i}∪{ i} ≤ ≤ | |
1 L (n 2)/2
≤| | ≤ ⌊ − ⌋
(vii) A′ B′ a b for each i satisfying 1 i n, and for each L such that L is even and
L∪ L∪{ i}∪{ i} ≤ ≤ | |
2 L (n 2)/2
≤| | ≤ ⌊ − ⌋
It can be seen that the number of hyperedges in H is (n+4)2n−3 +(n 2)m when n is odd
c
−
and (n+4)2n−3+(n 2)m +n n−2 /2+ n /2 when n is even. Mathews et al. [11] showed
− c (n−2)/2 n/2
that H is non-2-colorable, thereby establishing the following result on m(n).
(cid:0) (cid:1) (cid:0) (cid:1)
Lemma 4.
(n+4)2n−3+(n 2) m(n 2) if n is odd
m(n) − · −
≤ ((n+4)2n−2+(n 2) m(n 2)+n n−2 /2+ n /2 if n is even
− · − (n−2)/2 n/2
(cid:0) (cid:1) (cid:0) (cid:1)
This construction improved the upper bound on m(11) to m(11) 25449 and further improved
≤
theabove-mentioned upperboundson m(13) andm(17) tom(13) 297347 andm(17) 13201419,
≤ ≤
respectively.
6
3 Improved Upper Bounds for Small n
3.1 Generalized Abbott-Hanson-Toft Construction
For any k 1, we construct a non-2-colorable n-uniform hypergraph H = (V,E) for an integer
≥
n satisfying n > 2k. This construction uses a non-2-colorable (n 2k)-uniform hypergraph H =
c
−
(V ,E ) with E = e ,e ,...,e . Consider two disjoint sets of vertices A = a ,a ,...,a
c c c { 1 2 mc} { 1 2 n+k−1}
and B = b ,b ,...,b , each disjoint with V . Let us define to be the collection of all
1 2 n+k−1 c i
{ } I
i-element subsets of the set = 1,2,...,n+k 1 . For an I , consider K I. We
k−1 I
N { − } ∈ I ⊆ N \
define A = a , B = b , A = a , B = b , A = A (A A )
I i∈I{ i} I i∈I{ i} KI i∈KI{ i} KI i∈KI{ i} KI \ KI ∪ I
and B = B (B B ).
KI S\ KI ∪ I S S S
The non-2-colorable n-uniform hypergraph H = (V,E) is constructed with the vertex set V =
V A B. The hyperedge set E consists of the following hyperedges:
c
∪ ∪
(i) e A B for each I and all i satisfying 1 i m
i I I k c
∪ ∪ ∈ I ≤ ≤
(ii) A B for each K such that K is odd and 1 K n/2 , for each I
KI ∪ KI I | I| ≤ | I| ≤ ⌊ ⌋ ∈ Ik−1
(iii) A B for each K such that K is even and 0 K n/2 , for each I
KI ∪ KI I | I| ≤ | I|≤ ⌊ ⌋ ∈ Ik−1
The number of hyperedges in H is n+k−1 m + n+k−1 2n−1 when n is odd and n+k−1 m +
k c k−1 k c
n+k−1 2n−1+ n /2 when n is even.
k−1 n/2 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
For a 2-coloring of H, we call a ,b to be a matching pair if both the vertices are colored by
(cid:0) (cid:1)(cid:0) (cid:0) (cid:1) (cid:1) { i i}
the same color. For a given I , we define A to be the set blue vertices in A A , and
∈ Ik−1 blueI \ I
B = b :a A . Let A = A (A A ) and B = b : a A .
blueI { i i ∈ blueI} blueI \ I ∪ blueI blueI { i i ∈ blueI}
Lemma 5. Consider any I . If there is no matching pair of vertices between A A and
k−1 I
∈ I \
B B in a 2-coloring χ of hypergraph H, then there exists at least one monochromatic hyperedge
I
\
in the coloring χ.
Proof. Assume for the sake of contradiction that χ is a proper 2-coloring of hypergraph H with no
matching pair of vertices between A A and B B . We arrive at a contradiction in each of the
I I
\ \
cases below.
Case 1. 1 A n/2
≤| blueI|≤ ⌊ ⌋
If A is odd, the hyperedge A B is monochromatic in blue.
| blueI| blueI ∪ blueI
If A is even, the hyperedge A B is monochromatic in red.
| blueI| blueI ∪ blueI
Case 2. n/2 < A < n
⌊ ⌋ | blueI|
If n A is odd, the hyperedge A B is monochromatic in red.
−| blueI| blueI ∪ blueI
If n A is even, the hyperedge A B is monochromatic in blue.
−| blueI| blueI ∪ blueI
Case 3. A = 0 or A = n
| blueI| | blueI|
If A = 0, A A is monochromatic in red. If A = n, A A is monochromatic in
| blueI| \ I | blueI| \ I
blue.
7
Proof of Result 1. Let us assume for the sake of contradiction that there exists a proper 2-coloring
χ for hypergraph H. We know that the core hypergraph H is a non-2-colorable (n 2k)-uniform
c
−
hypergraph and thus has a monochromatic hyperedge in the coloring χ. Without loss of generality,
assume H to be monochromatic in red. The hyperedges added in Step (i) ensures that no more
c
than (k 1) matching pairs of red vertices exist in χ. This implies that there exists an I
k−1
− ∈ I
such that there is no matching pair of red vertices from A′ = A A and B′ = B B . As a result,
I I
\ \
it follows from Lemma 5 that there exists at least one matching pair of blue vertices from A′ and
B′. Let a ,b be such a matching pair of blue vertices, where a A′ and b B′. This leads to
p p p p
{ } ∈ ∈
a contradiction in each of the following cases.
Case 1. 1 A n/2
≤| blueI|≤ ⌈ ⌉
If A is odd, the hyperedge A B is monochromatic in blue.
| blueI| blueI ∪ blueI
If A is even, the hyperedge B b A a is monochromatic in blue.
| blueI| blueI ∪{ p}∪ blueI \{ p}
Case 2. n/2 < A < n
⌈ ⌉ | blueI|
Ifn A isodd, B b iseven. Therefore,thehyperedgeB b A a
−| blueI| | blueI∪{ p}| blueI∪{ p}∪ blueI\{ p}
is monochromatic in blue.
If n A is even, the hyperedge A B is monochromatic in blue.
−| blueI| blueI ∪ blueI
Case 3. A = n
| blueI|
If A = n, A′ is monochromatic in blue.
| blueI|
This completes the proof that H is non-2-colorable. Therefore, we arrive at the following result.
n+k−1 m(n 2k)+ n+k−1 2n−1 if n is odd
m(n) k − k−1
≤ ((cid:0)n+kk−1(cid:1)m(n−2k)+(cid:0)n+k−k−11(cid:1) 2n−1+ (nn−−2)2/2 if n is even
(cid:16) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
We set k = 2 in Result 1 and use m(9) 2401 from Table 1 to get an improvement of the upper
≤
bound on m(13) to m(13) 14 m(9)+ 14 212 275835.
≤ 2 1 ≤
(cid:0) (cid:1) (cid:0) (cid:1)
3.1.1 Optimization of m(n)
From theconstruction above, we obtain thefollowing forany integer ngreater thanalarge constant
n > 0.
0
n+k 1 n+k 1 n 2
m(n) − m(n 2k)+ − 2n−1+ −
≤ k − k 1 (n 2)/2
(cid:18) (cid:19) (cid:18) − (cid:19)(cid:16) (cid:18) − (cid:19)(cid:17)
n+k 1 n+k 1
− m(n 2k)+ − 2n
≤ k − k 1
(cid:18) (cid:19) (cid:18) − (cid:19)
Let k = np, where 1 p <0.5. Therefore,
n ≤
n+np 1 n+np 1
m(n) − m(n 2np)+ − 2n
≤ np − np 1
(cid:18) (cid:19) (cid:18) − (cid:19)
n+np n+np
m(n 2np)+ 2n.
≤ np − np
(cid:18) (cid:19) (cid:18) (cid:19)
8
Applying n < (en)k, we obtain the following.
k k
np
e(n+np)
(cid:0)m(cid:1)(n) < m(n 2np)+2n
np −
(cid:18) (cid:19)
(cid:16) (cid:17)
np (n−2np)p
e(1+p) e(1+p)
< 2n + m(n(1 2p)2)+2n−2np
p p −
" !#
(cid:18) (cid:19) (cid:18) (cid:19)
.
.
.
e(1+p) Psi=−01np(1−2p)i
< m(n(1 2p)s)
− p
(cid:18) (cid:19)
s−1 Pi np(1−2p)j
+ 2n(1−2p)i e(1+p) j=0
p
i=0 (cid:18) (cid:19)
X
n(1−(1−2p)s)
e(1+p) 2
= m(n(1 2p)s)
− p
(cid:18) (cid:19)
s−1 n(1−(1−2p)(i+1))
+ 2n(1−2p)i e(1+p) 2
p
i=0 (cid:18) (cid:19)
X
n
e(1+p) 2
m(n(1 2p)s)
≤ − p
(cid:18) (cid:19)
n s−1 n(1−2p)i
e(1+p) 2 2
+
p (1−2p)/2
e(1+p) !
(cid:18) (cid:19) i=0
X p
For any integer c > 0, we observe that there (cid:16)exists a(cid:17)constant c′ > lnn−lnc such that n(1 2p)s < c
2lnn −
for s c′nlnn. Using s= c′nlnn in the above equation, we have
≥
n c′nlnn−1 n(1−2p)i
e(1+p) 2 2
m(n) m(c)+ .
≤ p (1−2p)/2
" e(1+p) ! #
(cid:18) (cid:19) i=0
X p
−1 (cid:16) (cid:17)
e(1+p) (1−2p)/2
We observe that 2 increases if p increases and its value is less than 1 for 0 <
p
p 0.238. Using p(cid:16)= 0.238 in the a(cid:17)bove equation, we obtain
≤ (cid:0) (cid:1)
e(1+p) n
m(n) < 2 m(c)+c′nlnn
p
(cid:16) (cid:17)
= O(3.7596n n(cid:2)lnn) (cid:3)
·
= O(3.76n).
3.2 Multi-Core Construction
Consider an integer k satisfying 0 < k < n. We define w = n/k ,x = n mod k,y = k/x and
⌊ ⌋ ⌊ ⌋
z = k mod x. Amulti-core constructionmakes useofanon-2-colorable (n k)-uniformhypergraph
−
H = (V ,E ),atotalofwidenticalnon-2-colorablek-uniformhypergraphsH = (V ,E ),...,H =
c c c 1 1 1 w
(V ,E ) and a total of y identical non-2-colorable x-uniform hypergraphs H′ = (V′,E′),...,H′ =
w w 1 1 1 y
(V′,E′). The vertex sets of the hypergraphs H , H ,...,H , H′,...,H′ are pairwise disjoint.
y y c 1 w 1 y
Let us denote E = e ,e ,...,e , E = e1,e1,...,e1 , ..., E = ew,ew,...,ew , E′ =
c { 1 2 mc} 1 { 1 2 mk} w { 1 2 mk} 1
e′1,e′1,...,e′1 , ..., E′ = e′y,e′y,...,e′y . Consider a vertex set A = a ,a ,...,a ,
{ 1 2 mx} y { 1 2 mx} { 1 2 x+z−1}
disjoint with each of V , V ,...,V , V′,...,V′. We define as the collection of all p-element
c 1 w 1 y Ap
9
subsets of the vertex set A. Let = j j j : (j ,j ,...,j ) E E E and
1 2 w 1 2 w 1 2 w
E { ∪ ∪···∪ ∈ × ×···× }
′ = j′ j′ j′ :(j′,j′,...,j′) E′ E′ E′ .
E { 1 ∪ 2∪···∪ y 1 2 y ∈ 1× 2×···× y}
We define the construction of the non-2-colorable n-uniform hypergraph H = (V,E) as follows.
The vertex set is V = V A V V V′ V′. The construction of the hyperedges
c ∪ ∪ 1 ∪···∪ w ∪ 1 ∪···∪ y
belonging to E depends on the values of x and z as follows.
Case 1. For x > 0 and z > 0, E contains the following hyperedges.
(i) e el for every triple i,j,l satisfying 1 i m , 1 j m and 1 l w
i ∪ j ≤ ≤ c ≤ ≤ k ≤ ≤
′j
(ii) e e for every triple i,j,e satisfying 1 i m , 1 j y and e
i ∪ ≤ ≤ x ≤ ≤ ∈ E
(iii) e e′ S for every triple i,e,S satisfying 1 i m , e′ ′ and S
i c z
∪ ∪ ≤ ≤ ∈ E ∈ A
(iv) e S for every pair e,S satisfying e and S
x
∪ ∈E ∈ A
Case 2. For x > 0 and z = 0, E contains the following hyperedges.
(i) e el for every triple i,j,l satisfying 1 i m , 1 j m and 1 l w
i ∪ j ≤ ≤ c ≤ ≤ k ≤ ≤
′j
(ii) e e for every triple i,j,e satisfying 1 i m , 1 j y and e
i ∪ ≤ ≤ x ≤ ≤ ∈ E
(iii) e e′ for every pair i,e′ satisfying 1 i m and e′ ′
i c
∪ ≤ ≤ ∈ E
Case 3. For x = 0, E contains the following hyperedges.
(i) e el for every triple i,j,l satisfying 1 i m , 1 j m and 1 l w
i ∪ j ≤ ≤ c ≤ ≤ k ≤ ≤
(ii) e for each e
∈ E
The number of hyperedges in H is given by
wm m +ym (m )w + x+z−1 m (m )y + x+z−1 (m )w if x > 0,z > 0
c k x k z c x x k
E = wm m +ym (m )w +m (m )y if x > 0,z = 0
| |  c k x k (cid:0) c x(cid:1) (cid:0) (cid:1)
wm m +(m )w if x = 0
c k k

Proof of Result2. For the sake of contradiction, let us assume that χ is a proper 2-coloring of
H. Without loss of generality, let the hypergraph H be monochromatic in red in the coloring
c
χ. The hyperedges formed in Step (i) in each of the cases ensure that the hypergraphs H are
j
monochromatic in blue for each j 1,...,w .
∈ { }
Case 1. If x > 0 and z > 0, the hyperedges formed in Step (ii) ensure that the hypergraphs H′ are
l
monochromaticinredforeachl 1,2,...,y . Itcanbenotedfromthehyperedgesgenerated
∈ { }
in Step (iii) that there are at most z 1 red vertices in the set A. This implies that A has at
−
least x blue vertices. The hyperedges formed in Step (iv) ensure that there are at most x 1
−
blue vertices in A. Thus, we have a contradiction.
Case 2. If x > 0 and z = 0, the hyperedges formed in Step (ii) ensure that the hypergraphs H′ are
l
monochromatic in red for each l 1,2,...,y . It can be easily noted that the hyperedges
∈ { }
generatedinStep(iii)includearedmonochromatichyperedge. Thus,wehaveacontradiction.
Case 3. If x = 0, it immediately follows that we have a blue monochromatic hyperedge in the hyper-
edges generated by Step (ii) of the construction. This leads to a contradiction.
10