${\ell}$-Degree Turán Density

Author name not available (Why is that?)

Abstract: Let

H_{n}

be a

k

-graph on

n

vertices. For

0 l e e l l < k

and an

e l l

-subset

T

of

V (H_{n})

, define the degree

d e g (T)

of

T

to be the number of

(k - e l l)

-subsets~

S

such that

S c u p T

is an edge in~

H_{n}

. Let the minimum

e l l

-degree of

H_{n}

be

and

. Given a family

m a t h c a l F

of

k

-graphs, the

e l l

-degree Tur'an number

e x t {e x}_{e l l} (n, m a t h c a l F)

is the largest

d e l t a_{e l l} (H_{n})

over all

m a t h c a l F

-free

k

-graphs

H_{n}

on

n

vertices. Hence,

e x t {e x}_{0} (n, m a t h c a l F)

is the Tur'an number. We define

e l l

-degree Tur'an density to be

pi^k_{ell}(mathcal{F}) = limsup_{n ightarrow infty} frac{ ext{ex}_{ell}(n, mathcal{F} )}{ �inom{n- ell}{k}}.

In this paper, we show that for

k > e l l > 1

, the set of

p i_{e l l}^{k} (m a t h c a l F)

is dense in the interval

[0, 1)

. Hence, there is no "jump" for

e l l

-degree Tur'an density when

k > e l l > 1

. We also give a lower bound on

p i_{e l l}^{k} (m a t h c a l F)

in terms of an ordinary Tur'an density.

No records found.

No records found.