9 research outputs found
2-cancellative hypergraphs and codes
A family of sets F (and the corresponding family of 0-1 vectors) is called
t-cancellative if for all distict t+2 members A_1,... A_t and B,C from F the
union of A_1,..., A_t and B differs from the union of A_1, ..., A_t and C. Let
c(n,t) be the size of the largest t-cancellative family on n elements, and let
c_k(n,t) denote the largest k-uniform family. We significantly improve the
previous upper bounds, e.g., we show c(n,2) n_0). Using an
algebraic construction we show that the order of magnitude of c_{2k}(n,2) is
n^k for each k (when n goes to infinity).Comment: 20 page
Toward a density Corr\'{a}di--Hajnal theorem for degenerate hypergraphs
Given an -graph with , let denote
the maximum number of edges in an -vertex -graph with at most
pairwise vertex-disjoint copies of . Extending several old results and
complementing prior work [J. Hou, H. Li, X. Liu, L.-T. Yuan, and Y. Zhang. A
step towards a general density Corr\'{a}di--Hajnal theorem. arXiv:2302.09849,
2023.] on nondegenerate hypergraphs, we initiate a systematic study on
for degenerate hypergraphs . For a broad class of
degenerate hypergraphs , we present near-optimal upper bounds for
when is sufficiently large and lies in
intervals ,
,
and , where
is a constant depending only on . Our results reveal very
different structures for extremal constructions across the three intervals, and
we provide characterizations of extremal constructions within the first
interval. Additionally, for graphs, we offer a characterization of extremal
constructions within the second interval. Our proof for the first interval also
applies to a special class of nondegenerate hypergraphs, including those with
undetermined Tur\'{a}n densities, partially improving a result in [J. Hou, H.
Li, X. Liu, L.-T. Yuan, and Y. Zhang. A step towards a general density
Corr\'{a}di--Hajnal theorem. arXiv:2302.09849, 2023.]Comment: 37 pages, 4 figures, comments are welcom
A new construction for cancellative families of sets
we say a family, H, of subsets of a n-element set is cancellativ
A New Construction For Cancellative Families Of Sets
Following [2], we say a family, H, of subsets of a n-element set is cancellative if A[B = A[C implies B = C when A; B;C 2 H. We show how to construct cancellative families of sets with c2 :54797n elements. This improves the previous best bound c2 :52832n and falsifies conjectures of Erdős and Katona [3] and Bollobas [1]