38 research outputs found
Ramsey properties of randomly perturbed graphs: cliques and cycles
Given graphs , a graph is -Ramsey if for every
colouring of the edges of with red and blue, there is a red copy of
or a blue copy of . In this paper we investigate Ramsey questions in the
setting of randomly perturbed graphs: this is a random graph model introduced
by Bohman, Frieze and Martin in which one starts with a dense graph and then
adds a given number of random edges to it. The study of Ramsey properties of
randomly perturbed graphs was initiated by Krivelevich, Sudakov and Tetali in
2006; they determined how many random edges must be added to a dense graph to
ensure the resulting graph is with high probability -Ramsey (for
). They also raised the question of generalising this result to pairs
of graphs other than . We make significant progress on this
question, giving a precise solution in the case when and
where . Although we again show that one requires polynomially fewer
edges than in the purely random graph, our result shows that the problem in
this case is quite different to the -Ramsey question. Moreover, we
give bounds for the corresponding -Ramsey question; together with a
construction of Powierski this resolves the -Ramsey problem.
We also give a precise solution to the analogous question in the case when
both and are cycles. Additionally we consider the
corresponding multicolour problem. Our final result gives another
generalisation of the Krivelevich, Sudakov and Tetali result. Specifically, we
determine how many random edges must be added to a dense graph to ensure the
resulting graph is with high probability -Ramsey (for odd
and ).Comment: 24 pages + 12-page appendix; v2: cited independent work of Emil
Powierski, stated results for cliques in graphs of low positive density
separately (Theorem 1.6) for clarity; v3: author accepted manuscript, to
appear in CP
Almost-Fisher families
A classic theorem in combinatorial design theory is Fisher's inequality,
which states that a family of subsets of with all pairwise
intersections of size can have at most non-empty sets. One may
weaken the condition by requiring that for every set in , all but
at most of its pairwise intersections have size . We call such
families -almost -Fisher. Vu was the first to study the maximum
size of such families, proving that for the largest family has
sets, and characterising when equality is attained. We substantially refine his
result, showing how the size of the maximum family depends on . In
particular we prove that for small one essentially recovers Fisher's
bound. We also solve the next open case of and obtain the first
non-trivial upper bound for general .Comment: 27 pages (incluiding one appendix
Colouring set families without monochromatic k-chains
A coloured version of classic extremal problems dates back to Erd\H{o}s and
Rothschild, who in 1974 asked which -vertex graph has the maximum number of
2-edge-colourings without monochromatic triangles. They conjectured that the
answer is simply given by the largest triangle-free graph. Since then, this new
class of coloured extremal problems has been extensively studied by various
researchers. In this paper we pursue the Erd\H{o}s--Rothschild versions of
Sperner's Theorem, the classic result in extremal set theory on the size of the
largest antichain in the Boolean lattice, and Erd\H{o}s' extension to
-chain-free families.
Given a family of subsets of , we define an
-colouring of to be an -colouring of the sets without
any monochromatic -chains . We
prove that for sufficiently large in terms of , the largest
-chain-free families also maximise the number of -colourings. We also
show that the middle level, , maximises the
number of -colourings, and give asymptotic results on the maximum
possible number of -colourings whenever is divisible by three.Comment: 30 pages, final versio
Intersecting families of discrete structures are typically trivial
The study of intersecting structures is central to extremal combinatorics. A
family of permutations is \emph{-intersecting} if
any two permutations in agree on some indices, and is
\emph{trivial} if all permutations in agree on the same
indices. A -uniform hypergraph is \emph{-intersecting} if any two of its
edges have vertices in common, and \emph{trivial} if all its edges share
the same vertices.
The fundamental problem is to determine how large an intersecting family can
be. Ellis, Friedgut and Pilpel proved that for sufficiently large with
respect to , the largest -intersecting families in are the trivial
ones. The classic Erd\H{o}s--Ko--Rado theorem shows that the largest
-intersecting -uniform hypergraphs are also trivial when is large. We
determine the \emph{typical} structure of -intersecting families, extending
these results to show that almost all intersecting families are trivial. We
also obtain sparse analogues of these extremal results, showing that they hold
in random settings.
Our proofs use the Bollob\'as set-pairs inequality to bound the number of
maximal intersecting families, which can then be combined with known stability
theorems. We also obtain similar results for vector spaces.Comment: 19 pages. Update 1: better citation of the Gauy--H\`an--Oliveira
result. Update 2: corrected statement of the unpublished Hamm--Kahn result,
and slightly modified notation in Theorem 1.6 Update 3: new title, updated
citations, and some minor correction
Enumerating extensions of mutually orthogonal Latin squares
Two n×n Latin squares L1,L2 are said to be orthogonal if, for every ordered pair (x, y) of symbols, there are coordinates (i, j) such that L1(i,j)=x and L2(i,j)=y. A k-MOLS is a sequence of k pairwise-orthogonal Latin squares, and the existence and enumeration of these objects has attracted a great deal of attention. Recent work of Keevash and Luria provides, for all fixed k, log-asymptotically tight bounds on the number of k-MOLS. To study the situation when k grows with n, we bound the number of ways a k-MOLS can be extended to a (k+1)-MOLS. These bounds are again tight for constant k, and allow us to deduce upper bounds on the total number of k-MOLS for all k. These bounds are close to tight even for k linear in n, and readily generalise to the broader class of gerechte designs, which include Sudoku squares