35 research outputs found
Bio-based Products from Lignocellulosic Waste Biomass: A State of the Art
This review presents data on the chemical composition of harvest residues and food industry by-products as widely abundant representatives of lignocellulosic waste biomass. Pretreatment methods, with special emphasis on biological methods, are presented
as an important step in utilization of lignocellulosic waste biomass for the production of sustainable biofuels and high-value chemicals. Special attention was paid to the methods of lignin isolation and its possible utilization within lignocellulosic biorefinery. The objectives
of circular bioeconomy and the main aspects of lignocellulosic biorefinery are highlighted. Finally, current data on industrial, pilot, and research and development plants used in Europe for the production of a variety of bio-based products from different feedstocks are presented.
This work is licensed under a Creative Commons Attribution 4.0 International License
3āColor bipartite Ramsey number of cycles and paths
The k-colour bipartite Ramsey number of a bipartite graph H is the least integer n for which
every k-edge-coloured complete bipartite graph Kn,n contains a monochromatic copy of H. The
study of bipartite Ramsey numbers was initiated, over 40 years ago, by Faudree and Schelp and,
independently, by GyĀ“arfĀ“as and Lehel, who determined the 2-colour Ramsey number of paths. In
this paper we determine asymptotically the 3-colour bipartite Ramsey number of paths and (even)
cycles
Directed Ramsey number for trees
We call a family F of subsets of [n] s-saturated if it contains no s pairwise disjoint sets, and
moreover no set can be added to F while preserving this property (here [n] = {1, . . . , n}).
More than 40 years ago, ErdĖos and Kleitman conjectured that an s-saturated family of subsets
of [n] has size at least (1 ā 2
ā(sā1))2n. It is easy to show that every s-saturated family has size
at least 1
2
Ā· 2
n, but, as was mentioned by Frankl and Tokushige, even obtaining a slightly better
bound of (1/2 + Īµ)2n, for some fixed Īµ > 0, seems difficult. In this note, we prove such a result,
showing that every s-saturated family of subsets of [n] has size at least (1 ā 1/s)2n.
This lower bound is a consequence of a multipartite version of the problem, in which we seek a
lower bound on |F1| + . . . + |Fs| where F1, . . . , Fs are families of subsets of [n], such that there
are no s pairwise disjoint sets, one from each family Fi
, and furthermore no set can be added to
any of the families while preserving this property. We show that |F1| + . . . + |Fs| ā„ (s ā 1) Ā· 2
n,
which is tight e.g. by taking F1 to be empty, and letting the remaining families be the families
of all subsets of [n]
Large cliques and independent sets all over the place
We study the following question raised by Erd\H{o}s and Hajnal in the early
90's. Over all -vertex graphs what is the smallest possible value of
for which any vertices of contain both a clique and an independent set
of size ? We construct examples showing that is at most
obtaining a twofold sub-polynomial
improvement over the upper bound of about coming from the natural
guess, the random graph. Our (probabilistic) construction gives rise to new
examples of Ramsey graphs, which while having no very large homogenous subsets
contain both cliques and independent sets of size in any small subset
of vertices. This is very far from being true in random graphs. Our proofs are
based on an interplay between taking lexicographic products and using
randomness.Comment: 12 page
Clique minors in graphs with a forbidden subgraph
The classical Hadwiger conjecture dating back to 1940's states that any graph
of chromatic number at least has the clique of order as a minor.
Hadwiger's conjecture is an example of a well studied class of problems asking
how large a clique minor one can guarantee in a graph with certain
restrictions. One problem of this type asks what is the largest size of a
clique minor in a graph on vertices of independence number at
most . If true Hadwiger's conjecture would imply the existence of a clique
minor of order . Results of Kuhn and Osthus and Krivelevich and
Sudakov imply that if one assumes in addition that is -free for some
bipartite graph then one can find a polynomially larger clique minor. This
has recently been extended to triangle free graphs by Dvo\v{r}\'ak and
Yepremyan, answering a question of Norin. We complete the picture and show that
the same is true for arbitrary graph , answering a question of Dvo\v{r}\'ak
and Yepremyan. In particular, we show that any -free graph has a clique
minor of order , for some constant
depending only on . The exponent in this result is tight up to a
constant factor in front of the term.Comment: 11 pages, 1 figur
Halfway to Rotaās Basis Conjecture
In 1989, Rota made the following conjecture. Given n bases B1, . . . , Bn in an n-dimensional
vector space V , one can always find n disjoint bases of V , each containing exactly one element
from each Bi (we call such bases transversal bases). Rotaās basis conjecture remains wide open
despite its apparent simplicity and the efforts of many researchers (for example, the conjecture
was recently the subject of the collaborative āPolymathā project). In this paper we prove that
one can always find (1/2 ā o(1))n disjoint transversal bases, improving on the previous best
bound of ā¦(n/ log n). Our results also apply to the more general setting of matroids
Monochromatic trees in random tournaments
We prove that, with high probability, in every 2-edge-colouring of the random tournament on n vertices there is a monochromatic copy of every oriented tree of order O(n/ \sqrt{log n}. This generalizes a result of the first, third and fourth authors, who proved the same statement for paths, and is tight up to a constant factor
Minimum saturated families of sets
We call a family F of subsets of [n] s-saturated if it contains no s pairwise disjoint sets, and
moreover no set can be added to F while preserving this property (here [n] = {1, . . . , n}).
More than 40 years ago, ErdĖos and Kleitman conjectured that an s-saturated family of subsets
of [n] has size at least (1 ā 2
ā(sā1))2n. It is easy to show that every s-saturated family has size
at least 1
2
Ā· 2
n, but, as was mentioned by Frankl and Tokushige, even obtaining a slightly better
bound of (1/2 + Īµ)2n, for some fixed Īµ > 0, seems difficult. In this note, we prove such a result,
showing that every s-saturated family of subsets of [n] has size at least (1 ā 1/s)2n.
This lower bound is a consequence of a multipartite version of the problem, in which we seek a
lower bound on |F1| + . . . + |Fs| where F1, . . . , Fs are families of subsets of [n], such that there
are no s pairwise disjoint sets, one from each family Fi
, and furthermore no set can be added to
any of the families while preserving this property. We show that |F1| + . . . + |Fs| ā„ (s ā 1) Ā· 2
n,
which is tight e.g. by taking F1 to be empty, and letting the remaining families be the families
of all subsets of [n]