18 research outputs found
The induced saturation problem for posets
For a fixed poset , a family of subsets of is induced
-saturated if does not contain an induced copy of , but for
every subset of such that , then is an
induced subposet of . The size of the smallest such
family is denoted by . Keszegh, Lemons,
Martin, P\'alv\"olgyi and Patk\'os [Journal of Combinatorial Theory Series A,
2021] proved that there is a dichotomy of behaviour for this parameter: given
any poset , either or . In this paper we improve this general result showing that either
or . Our proof
makes use of a Tur\'an-type result for digraphs.
Curiously, it remains open as to whether our result is essentially best
possible or not. On the one hand, a conjecture of Ivan states that for the
so-called diamond poset we have ; so if true this conjecture implies our result is tight up to a
multiplicative constant. On the other hand, a conjecture of Keszegh, Lemons,
Martin, P\'alv\"olgyi and Patk\'os states that given any poset , either
or . We prove that this
latter conjecture is true for a certain class of posets .Comment: 12 page
-cross -intersecting families via necessary intersection points
Given integers and we call families
-cross
-intersecting if for all , , we have
. We obtain a strong generalisation of
the classic Hilton-Milner theorem on cross intersecting families. In
particular, we determine the maximum of for -cross -intersecting families in the
cases when these are -uniform families or arbitrary subfamilies of
. Only some special cases of these results had been proved
before. We obtain the aforementioned theorems as instances of a more general
result that considers measures of -cross -intersecting families. This
also provides the maximum of for
families of possibly mixed uniformities .Comment: 13 page
On oriented cycles in randomly perturbed digraphs
In 2003, Bohman, Frieze, and Martin initiated the study of randomly perturbed
graphs and digraphs. For digraphs, they showed that for every , there
exists a constant such that for every -vertex digraph of minimum
semi-degree at least , if one adds random edges then
asymptotically almost surely the resulting digraph contains a consistently
oriented Hamilton cycle. We generalize their result, showing that the
hypothesis of this theorem actually asymptotically almost surely ensures the
existence of every orientation of a cycle of every possible length,
simultaneously. Moreover, we prove that we can relax the minimum semi-degree
condition to a minimum total degree condition when considering orientations of
a cycle that do not contain a large number of vertices of indegree . Our
proofs make use of a variant of an absorbing method of Montgomery.Comment: 24 pages, 7 figures. Author accepted manuscript, to appear in
Combinatorics, Probability and Computin
Números de Turán en coloreos promedio para grafos completos
Ingeniero Civil MatemáticoUn coloreo de aristas de un grafo se llama γ-promedio si es que el número promedio de colores incidentes a cada vértice es a lo más γ. Dados n, m enteros positivos y γ un real positivo, el número de Turán promedio coloreado T(n, K_m, γ-promedio) corresponde a la máxima cantidad de aristas que puede tener un grafo de n vértices de manera que exista un coloreo γ-promedio que no contenga ninguna copia monocromática de K_m. Esta noción fue introducida por Caro, quien observa que la expansión (blow-up) de un grafo completo γ-promedio coloreado sin copias monocromáticas de K_m también es γ-promedio coloreado y tampoco posee copias monocromáticas de K_m. Con ello, Caro se pregunta de si el máximo de aristas buscado se alcanza en la expansión de un grafo completo γ-promedio coloreado que maximice el número de vértices bajo la condición de no contener una copia monocromática de K_m (un coloreo extremal para el número de Ramsey γ-promedio).
Yuster probó que la respuesta es afirmativa para el caso m=3 y γ=2, y además conjeturó que la respuesta es siempre afirmativa para todos los γ = ℓ ∈ N. En la presente memoria se demuestra esta conjetura de Yuster cuando m >= ℓ(ℓ+1)+1. Por otro lado, se demuestra también que la respuesta a la pregunta de Caro es negativa para un conjunto no numerable de valores de γ ∉ N
Hypergraphs with arbitrarily small codegree Tur\'an density
Let . Given a -uniform hypergraph , the minimum codegree
is the largest such that every -set of
is contained in at least edges. Given a -uniform hypergraph ,
the codegree Tur\'an density of is the smallest such that every -uniform hypergraph on vertices with
contains a copy of . Similarly as other
variants of the hypergraph Tur\'an problem, determining the codegree Tur\'an
density of a hypergraph is in general notoriously difficult and only few
results are known.
In this work, we show that for every , there is a -uniform
hypergraph with . This is in contrast to the
classical Tur\'an density, which cannot take any value in the interval
due to a fundamental result by Erd\H{o}s.Comment: 12 page
Cycle decompositions in 3-uniform hypergraphs
We show that -graphs on vertices whose codegree is at least can be decomposed into tight cycles and admit Euler tours, subject to
the trivial necessary divisibility conditions. We also provide a construction
showing that our bounds are best possible up to the term. All together,
our results answer in the negative some recent questions of Glock, Joos,
K\"uhn, and Osthus.Comment: 28 pages, fixed small errors in references and compilatio