226 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
On the use of senders for minimal Ramsey theory
This thesis investigates problems related to extremal and probabilistic graph theory. Our focus lies on the highly dynamic field of Ramsey theory. The foundational result of this field was proved in 1930 by Franck P. Ramsey. It implies that for every integer t and every sufficiently large complete graph Kn, every colouring of the edges of Kn with colours red and blue contains a red copy or a blue copy of Kt.
Let q â©Ÿ 2 represent a number of colours, and let H1,..., Hq be graphs. A graph G is said to be q-Ramsey for the tuple (H1,...,Hq) if, for every colouring of the edges of G with colours {1, . . . , q}, there exists a colour i and a monochromatic copy of Hi in colour i. As we often want to understand the structural properties of the collection of graphs that are q-Ramsey for (H1,..., Hq), we restrict our attention to the graphs that are minimal for this property, with respect to subgraph inclusion. Such graphs are said to be q-Ramsey-minimal for (H1,..., Hq).
In 1976, Burr, ErdĆs, and LovĂĄsz determined, for every s, t â©Ÿ 3, the smallest minimum degree of a graph G that is 2-Ramsey-minimal for (Ks, Kt). Significant efforts have been dedicated to generalising this result to a higher number of colours, qâ©Ÿ3, starting with the âsymmetricâ q-tuple (Kt,..., Kt). In this thesis, we improve on the best known bounds for this parameter, providing state-of-the-art bounds in different (q, t)-regimes. These improvements rely on constructions based on finite geometry, which are then used to prove the existence of extremal graphs with certain key properties. Another crucial ingredient in these proofs is the existence of gadget graphs, called signal senders, that were initially developed by Burr, ErdĆs, and LovĂĄsz in 1976 for pairs of complete graphs. Until now, these senders have been shown to
exist only in the two-colour setting, when q = 2, or in the symmetric multicolour setting, when H1,..., Hq are pairwise isomorphic. In this thesis, we then construct similar gadgets for all tuples of complete graphs, providing the first known constructions of these tools in the multicolour asymmetric setting. We use these new senders to prove far-reaching generalisations of several classical results in the area
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
On the structure of graphs with forbidden induced substructures
One of the central goals in extremal combinatorics is to understand how the global structure of a combinatorial object, e.g. a graph, hypergraph or set system, is affected by local constraints.
In this thesis we are concerned with structural properties of graphs and hypergraphs which locally do not look like some type of forbidden induced pattern. Patterns can be single subgraphs, families of subgraphs, or in the multicolour version colourings or families of colourings of subgraphs.
ErdĆs and Szekeres\u27s quantitative version of Ramsey\u27s theorem asserts that in every -edge-colouring of the complete graph on vertices there is a monochromatic clique on at least vertices. The famous ErdĆs-Hajnal conjecture asserts that forbidding fixed colourings on subgraphs ensures much larger monochromatic cliques. The conjecture is open in general, though a few partial results are known. The first part of this thesis will be concerned with different variants of this conjecture: A bipartite variant, a multicolour variant, and an order-size variant for hypergraphs.
In the second part of this thesis we focus more on order-size pairs; an order-size pair is the family consisting of all graphs of order and size , i.e. on vertices with edges. We consider order-size pairs in different settings: The graph setting, the bipartite setting and the hypergraph setting. In all these settings we investigate the existence of absolutely avoidable pairs, i.e. fixed pairs that are avoided by all order-size pairs with sufficiently large order, and also forcing densities of order-size pairs , i.e. for approaching infinity, the limit superior of the fraction of all possible sizes , such that the order-size pair does not avoid the pair
The existence of subspace designs
We prove the existence of subspace designs with any given parameters,
provided that the dimension of the underlying space is sufficiently large in
terms of the other parameters of the design and satisfies the obvious necessary
divisibility conditions. This settles an open problem from the 1970s. Moreover,
we also obtain an approximate formula for the number of such designs.Comment: 61 page
Graphs of low average degree without independent transversals
An independent transversal of a graph G with a vertex partition P is an independent set of G intersecting each block of P in a single vertex. Wanless and Wood proved that if each block of P has size at least t and the average degree of vertices in each block is at most t/4, then an independent transversal of P exists. We present a construction showing that this result is optimal: for any Δ>0 and sufficiently large t, there is a family of forests with vertex partitions whose block size is at least t, average degree of vertices in each block is at most (1/4+Δ)t, and there is no independent transversal. This unexpectedly shows that methods related to entropy compression such as the Rosenfeld-Wanless-Wood scheme or the Local Cut Lemma are tight for this problem. Further constructions are given for variants of the problem, including the hypergraph version
Abundance: Asymmetric Graph Removal Lemmas and Integer Solutions to Linear Equations
We prove that a large family of pairs of graphs satisfy a polynomial
dependence in asymmetric graph removal lemmas. In particular, we give an
unexpected answer to a question of Gishboliner, Shapira, and Wigderson by
showing that for every , there are -abundant graphs of
chromatic number . Using similar methods, we also extend work of Ruzsa by
proving that a set which avoids solutions
with distinct integers to an equation of genus at least two has size
. The best previous bound was and the
exponent of is best possible in such a result. Finally, we investigate
the relationship between polynomial dependencies in asymmetric removal lemmas
and the problem of avoiding integer solutions to equations. The results suggest
a potentially deep correspondence. Many open questions remain.Comment: 28 pages, 4 figure
On The Random Tur\'an number of linear cycles
Given two -uniform hypergraphs and the Tur\'an number is the maximum number of edges in an -free subgraph of . We study the
typical value of when , the Erd\H{o}s-R\'enyi
random -uniform hypergraph, and , the -uniform linear
cycle of length . The case of graphs () is a longstanding open
problem that has been investigated by many researchers. We determine the order
of magnitude of for all
and all up to polylogarithmic factors for all values of
.
Our proof is based on the container method and uses a balanced
supersaturation result for linear even cycles which improves upon previous such
results by Ferber-Mckinley-Samotij and Balogh-Narayanan-Skokan.Comment: 16 pages. arXiv admin note: substantial text overlap with
arXiv:2007.1032
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