On the extremal families for the Kruskal–Katona theorem

by   Oriol Serra, et al.

In <cit.>, the authors have shown a characterization of the extremal families for the Kruskal-Katona Theorem. We further develop some of the arguments given in <cit.> and give additional properties of these extremal families. Füredi-Griggs/Mörs theorem from 1986/85 <cit.> claims that, for some cardinalities, the initial segment of the colexicographical is the unique extremal family; we extend their result as follows: the number of (non-isomorphic) extremal families strictly grows with the gap between the last two coefficients of the k-binomial decomposition. We also show that every family is an induced subfamily of an extremal family, and that, somewhat going in the opposite direction, every extremal family is close to being the inital segment of the colex order; namely, if the family is extremal, then after performing t lower shadows, with t=O(log(log n)), we obtain the initial segment of the colexicographical order. We also give a “fast” algorithm to determine whether, for a given t and m, there exists an extremal family of size m for which its t-th lower shadow is not yet the initial segment in the colexicographical order. As a byproduct of these arguments, we give yet another characterization of the families of k-sets satisfying equality in the Kruskal–Katona theorem. Such characterization is, at first glance, less appealing than the one in <cit.>, since the additional information that it provides is indirect. However, the arguments used to prove such characterization provide additional insight on the structure of the extremal families themselves.


page 1

page 2

page 3

page 4


Extremal families for the Kruskal–Katona theorem

Given a family S of k–subsets of [n], its lower shadow Δ(S) is the famil...

On Restricted Intersections and the Sunflower Problem

A sunflower with r petals is a collection of r sets over a ground set X ...

r-wise fractional L-intersecting family

Let L = {a_1/b_1, ... , a_s/b_s}, where for every i ∈ [s], a_i/b_i∈ [0,1...

Short rainbow cycles for families of matchings and triangles

A generalization of the famous Caccetta–Häggkvist conjecture, suggested ...

Modular and fractional L-intersecting families of vector spaces

In the first part of this paper, we prove a theorem which is the q-analo...

On a characterization of exponential and double exponential distributions

Recently, G. Yanev obtained a characterization of the exponential family...

𝒮-adic characterization of minimal ternary dendric subshifts

Dendric subshifts are defined by combinatorial restrictions of the exten...

Please sign up or login with your details

Forgot password? Click here to reset