On hypergraph cliques with chromatic number 3 and a given number of vertices Текст научной статьи по специальности «Математика»

УДК 519.179.1 + 519.174

D. D. Cherkashin1, A. B. Kulikov1, A. M. Raigorodskii2,3,4

1 Saint-Petersbourg State University, Mathematics and Mechanics Faculty;

2 Moscow State University, Mechanics and Mathematics Faculty, Department of

Mathematical Statistics;

3 Moscow Institute of Physics and Technology, Faculty of Innovations and High Technology,

Department of Discrete Mathematics;

4 Yandex Research Division

On hypergraph cliques with chromatic number 3 and a given number of vertices

In 1973, P. Erdos and L. Lovasz pointed out that any hypergraph with pairwise intersecting edges has chromatic number 2 or 3. In the first case, this hypergraph can have any number of edges. However, Erdos and Lovasz proved that in the second case, the number of edges is bounded from above. For example, if a hypergraph is n-uniform, has pairwise intersecting edges and chromatic number 3, the number of its edges is less than nn. Recently, D.D. Cherkashin improved this bound (see [2]).

In this paper, we further improve it, when the number of vertices of an n-uniform hypergraph is bounded from above by the value nm with some m = m(n).

Keywords: hypergraph clique, chromatic number.

1. Introduction and the main result

This paper is devoted to a problem in extremal hypergraph theory, which goes back to P. Erdos and L. Lovasz (see [3]). Before giving the exact statement of the problem, we recall some definitions and introduce some notation.

Let H = (V,E) be a hypergraph without multiple edges. We call it n-uniform if any of its edges has cardinality n: for every e G E, we have |e| = n. The chromatic number of the hypergraph H = (V, E) is the minimum number x(H) of colors needed to color all the vertices in V, so that any edge e G E contains at least two vertices of some different colors. Finally, a

hypergraph is said to form a clique if its edges are pairwise intersecting.

In 1973, Erdos and Lovasz pointed out that if an n-uniform hypergraph H = (V, E) forms a clique, then x(H) G {2, 3}. They also observed that in the case x(H) = 3, one is sure to have

|E| ^ nn (see [3]). Thus, the following definition is justified:

M(n) = max{|E| : 3 an n — uniform clique H = (V, E) with x(H) = 3}.

Obviously, this definition has no meaning for x(H) = 2.

Theorem 1 (P. Erdos, L. Lovasz, [3]). The inequalities hold

n! (i + i + ... + i) < M(n) < n".

Almost nothing better is done in the past 35 years. In [5], the estimate M(n) ^ (l — nn

is mentioned as “to appear”. However, we fail to find the corresponding paper.

This work was supported by the grant of RFBR N 12-01-00683, the grant of the Russian President N MD-

666.2012.1, and the grant NSh-2519.2012.1.

At the same time, another value r(n) is introduced in [6]:

r(n) = max{|E| : 3 an n — uniform clique H = (V, E) s.t. t(H) = n},

where t(H) is the covering number of H, i.e.,

t(H) = min{|f | : f C V, V e £ E f if e = 0}.

Clearly, for any n-uniform clique H, we have t(H) ^ n (since every edge forms a cover), and if x(H) = 3, then t(H) = n. Hence, M(n) ^ r(n). Lovasz pointed out that, for r(n), the same estimates, as in Theorem 1, apply and conjectured that the lower estimate is best possible. In 1996, P. Frankl, K. Ota, and N. Tokushige (see [4]) disproved this conjecture and showed that

(\ n— 1 f) '

In [2], D.D. Cherkashin discovered a new upper bound on the initial value of M(n), which is actually true for r(n) as well.

Theorem 2 (D.D. Cherkashin, [2]). There exists a constant c > 0 such that

M(n) ^ cnu~ 2 In n.

To present the main result of this paper we take any natural numbers n, m ^ 2, and put

q(n, m)

n

2m

A(n, m) =

2q{n,m) /nm

i=0

We note that

A(n, m) ^ ^ + 1

where

n

en

2q(n,m)/ \m

A'(n, m) = ( ^ + 1

n/m

2q(n, m)

n/m

. m > V 2q(n, m) Obviously, if m is a function of n, which is o(n) as n ^ to, then

A/(n, m) =

m

nn ■ A/(n, m),

nw(n) ’

where w(n) ^ to as n ^ to. Hence, A(n,m) = o (mnn-1).

Theorem 3. Let m ^ 2 be any function of n £ N which is o(n) as n ^ to; moreover, m(n) ^ -For any n ^ 4 and any n-uniform clique H = (V, E) with x(H) = 3 and

|V| ^ nm(n), we have

|E| ^ 4m(n)nn 1 + A(n, m(n)) = (4 + o(1))m(n)n

n1

e

Clearly, if m(n) ^ Cy/nlnn with some constant c > 0, the bound in Theorem 3 is stronger than that in Theorem 2. Note that the number of vertices in any n-uniform clique with chromatic number 3 is less than 4n (see [3]). Unfortunately, n^rlXnn = e0<<a\ so that Theorem 3 does not cover all possible values of |V|.

2. Proof of Theorem 3

Let us fix n ^ 4 and put m = m(n), q = q(n,m), A = A(n,m). Fix an n-uniform clique H = (V, E) with x(H) = 3 and | V| ^ nm. For any set W C V, we denote by E(W) the set of all edges B £ E such that W C B. Also, we denote by EW the set of all edges B £ E such that W if B = 0. Clearly, E(W) C EW. Let

Q = {1, 2, 3,...,q}U{n — q +1,n — q + 2,..., n}.

The two parts of the set Q do not intersect and do not cover the whole set {1,... ,n}, for to ^ 2. Moreover, Q is not empty, for to ^ hence q ^ 1.

Lemma 1. Let W C V, i = \W\. Either there exists a vertex x G W such that deg x ^ or there exist two edges B1, B2 £ E such that B1, B2 £ EW and |B1 f B2| £ Q.

Proof of Lemma 1. If there exists a vertex x E W such that deg x ^ -—k---------------, we are done.

If there are no such vertices, then

|Ew| ^ ^ deg x < |E| — A.

x€W

Hence, |E \ EW| > A. We are to show that there exist two edges B1,B2 £ E \ EW with |B1 f B2| £ Q. Suppose to the contrary that for any B1, B2 £ E \ EW, we have |B1 f B2| £ Q. We further prove that |E \ EW| ^ A. We find a contradiction and thus complete the proof of Lemma 1.

In principle, we can only cite [8]. Instead, we use a version of the linear algebra method in combinatorics (see [1] and [7]). To any edge B from E \ EW we assign the vector x = = (x1,... , xv) £ {0,1}v, where v = |V| ^ nm and xv = 1, if and only if v £ B. In particular, x1 + ... + xv = n. Let E \ EW ^ {x1,..., xs}.

We denote by (x, y) the Euclidean inner product of vectors x, y. Note that if B, B' £ E\EW and x, x' are the corresponding vectors, then |B f B'| = (x, x').

We take an arbitrary vector xv, v £ {1,..., s}, and consider the polynomial

FXv (y) = n (j — (xv , y)) £ R[y1,...,yv ].

j&Q\{n}

Finally, we get s polynomials FX1,..., FXs. All of them depend on v variables and their degree is less than |Q| ^ 2q. Certainly, any such polynomial is a linear combination of some monomials of the type

1, ■ ... ■ ,..., aVr ^ 1, avi + ... + aVr ^ |Q| ^ 2q.

We replace each monomial of this type by yvi ■... ■ yVr and denote by FX1,..., FXs the resulting polynomials. They also depend on v variables and their degree is less than |Q| ^ 2q. Moreover, they span a linear space whose dimension is less than or equal to

Simultaneously, F'v(y) = FXv(y), provided that y £ {0,1}v and v £ {1,..., s}.

To show that s = |E \ EW | ^ A (needed to complete the proof) it suffices to establish the linear independence of the polynomials FXX ,..., FXs over R. Suppose that

c1Fx i (y) + ... + csFx s (y) = 0 Let y = xv, v £ {1,..., s}. Then (xv, y) = (xv, xv) = n and

FXv (y) = FXv (y) = FXv (xv) = 0.

However, if ^ = v, then (xM,y) = (xM,xv) £ Q \ {n}, i.e.

FxM (y) = FXM (y) = Fxm (xv) = 0.

Hence, cv = 0 for every v. Lemma 1 is proved.

Lemma 2. Let W C V, i = |W|, j = |E(W)|. Suppose that there exist two edges B1,B2 £

1

Am

E E \ Ew such that \Bi n B2\ £ Q. We put r = 1 + j—. Either there exists x £ W such that

2

\E(W U {x})| ^ or there exist x,y <0 W such that \E(W U {x, y})\ ^

n n2

Proof of Lemma 2. Let l = |B1 f B2| £ Q. Consider the set E(W). Since H is a clique,

any edge B £ E(W) intersects both B1 and B2. Either B intersects the set B1 f B2, or it has

common vertices with both B1 \ (B1 f B2) and B2 \ (B1 f B2). We denote by E1 the set of edges

of the first type; E2 = E(W) \ E1. By pigeon-hole principle, there is x £ B1 f B2 such that x

e |

belongs to at least —-j— edges from E\\ there are also ig5i\ (Bi n £>2) and y E B2\ (Bi n £>2)

|e2|

such that the set {x, y} belongs to at least ---------—j edges from E2. We are to show that for any

(n — l )

partition E(W) = E1 U E2, we have

either M > or ^ > -111

I ^ TV’ OT („_*)

which is equivalent to

|E1|2 |E2|

Here the worst case is that of 0;0 = —----------------------77. Let a = \EA. Then \E2\ = j — a and we

j2l 2 j(n — l )2

a2 j --- a

have -77-7 = —----------77. Solving this equation, we get

j2l 2 j(n — l )2

j72 + a/ (jl2)2 + 4 j2l2(n — I)2

a =

2(n — l)2

Of course, the value of |E1| (which is an integer) can differ from the real number a. However, we do know that

, lFi|2 If2| 1 ^ a2 max < —r-7-, —------—r- > ^

j2l2 ’ j(n — l)2 j j212'

Thus, we are to prove that -77 ^ , or ^ ^ r. We have

’ Jl n Jl

an I + \A2 + ^(n 02 in I f in \ n2 ^ 1 i In

jl 2(n — l)2 2(n — l)2 V \2(n — l)2/ (n — l)2 2(n — l)2'

The function , ^n is monotone increasing at I. Since I & Q, we can use the bound 2(n — l)2

I ^ 2^- Consequently,

an > i , In > i |_____________n2 > i |_______]_ _

jl ^ 2(n — I)2 ^ Am(n — I)2 ^ Am T'

Lemma 2 is proved.

Completion of the proof of Theorem 3. Let

|E|— A

k = min < |W| : W C V, 3 x £ W deg x ^

|W |

The value k is well-defined. Indeed, we take any edge W £ E. Since H is a clique, W intersects

IE \E ______ a

all the edges from E and so there exists x G W with deg x ^ —.

Let W0 be a set on which the value of k is reached. We take a vertex x £ W0 that has deg x ^ -—L—_ xhe latter inequality can be rewritten as |£'({x})| ^ -—L—. If fc ^ 2, we

can apply Lemmas 1 and 2 to W = {x}. Thus, we obtain either some set W' of two elements

\E\ a \E\ A 2

with \E(W')\ ^ ^^ or some set W" of three elements with \E(W")\ ^ ^

|E| — A T k—1

We continue this process until we get some set W with \W\ = k and ^ —t— • k_1

k n

(even if k = 1, we do have such a set).

In [3], Erdos and Lovasz prove that for any n-uniform clique, H = (V, E) with chromatic number 3, if W C V is of cardinality k, then |E(W)| ^ nn—k. In our case, we have |E| - A Tk—1

----^ nu . Therefore,

k nk—1

k— 1 n— 1

\E\^k.n-k.^ZT + A = k^ZT + A.

k

To complete the proof of Theorem 3 we are to show that for any k, ^ Am. It is easy

k T to see that the maximum value of —- is obtained for k = Am.

T k—1

3. Refinement of Theorem 3

For m = 2, one can prove a simple result that is, however, much better than that of Theorem 3.

Theorem 4. Let H = (V, E) be any n-uniform clique with x(H) = 3. We put v = |V|. Suppose n2

that v ^ —; where c can be any function of n such that c{n) G (1, n). We now put

--i

a = ceec .

Then

\E\ < (1 + o(l))^-(n/d)'\

If c is a constant, we get an exponential improvement for the Erdos and Lovasz bound, which is equal to nn. Otherwise, the improvement is even much larger.

Proof of Theorem 4. We take an arbitrary integer a £ (1,n) and consider all a-element subsets of V. The number of such subsets is Q). On the one hand, any edge from E contains exactly (n) subsets. On the other hand, any subset is enclosed in nn—a edges at most (see

[3]). Therefore, the number of edges does not exceed the quantity

n

(П)

To estimate this

quantity we use the bound

^ -^y and the Stirling formula. Hence,

n

n—a(v\

О

n+a

We now put

a =

a!

1 - — )n

ec

ng nl (n—a)\

+ 1.

Then n — a ^ —, so that

- ec

n!

л/2ттп №■)

(n — a)\ ^2тг(n - a) (^)

ana

and

|E| « (1+ o(1))

n

n+a

с“л/ёс • na(e2c)n~ae~n

(1+ o(1))

n

n

л/ёс • cne~ne2n~2a ,e3/2

—2 = (1 + 0(1)) t-(n/dy ec ■ c'le~'le Vе

^ (1 + o(1))

Theorem 4 is proved.

Note that for constant values of c, the choice of a in the proof is nearly optimal.

a

n

a

n

References

1. L. Babai, P. Frankl. Linear algebra methods in combinatorics. Part 1. — Department of Computer Science, The University of Chicago, Preliminary version 2, September 1992.

2. D. D. Cherkashin. On hypergraph cliques with chromatic number 3 // Moscow J. of Combinatorics and Number Th. —2011. —V. 1. —N. 3. —P. 3—11.

3. P. Erdos, L. Lovasz. Problems and results on 3-chromatic hypergraphs and some related questions // Infinite and Finite Sets, Colloquia Mathematica Societatis Janos Bolyai, North Holland. — 1975. — V. 10. — P. 609—627.

4. P. Frankl, K. Ota, N. Tokushige. Covers in uniform intersecting families and a counterexample to a conjecture of Lovasz // Journal of Combin. Th., Ser. A. — 1996. — V. 74.— P. 33-42.

5. T. Jensen, B. Toft. Graph coloring problems. — New York: Wiley Interscience, 1995.

6. L. Lovasz. On minimax theorems of combinatorics // Math. Lapok. — 1975. — V. 26.— P. 209—264 (in Hungarian).

7. A. M. Raigorodskii. The linear algebra method in combinatorics. — Moscow Centre for Continuous Mathematical Education (MCCME), Moscow, Russia, 2007 (book in Russian).

8. D. K. Ray-Chaudhury, R.M. Wilson. On t-designs // Osaka J. Math. — 1975. — V. 12.— P. 735-744.

Поступила в редакцию 10.09.2011