569 research outputs found
On the minimum degree of minimal Ramsey graphs for multiple colours
A graph G is r-Ramsey for a graph H, denoted by G\rightarrow (H)_r, if every
r-colouring of the edges of G contains a monochromatic copy of H. The graph G
is called r-Ramsey-minimal for H if it is r-Ramsey for H but no proper subgraph
of G possesses this property. Let s_r(H) denote the smallest minimum degree of
G over all graphs G that are r-Ramsey-minimal for H. The study of the parameter
s_2 was initiated by Burr, Erd\H{o}s, and Lov\'{a}sz in 1976 when they showed
that for the clique s_2(K_k)=(k-1)^2. In this paper, we study the dependency of
s_r(K_k) on r and show that, under the condition that k is constant, s_r(K_k) =
r^2 polylog r. We also give an upper bound on s_r(K_k) which is polynomial in
both r and k, and we determine s_r(K_3) up to a factor of log r
Strongly Refuting Random CSPs Below the Spectral Threshold
Random constraint satisfaction problems (CSPs) are known to exhibit threshold
phenomena: given a uniformly random instance of a CSP with variables and
clauses, there is a value of beyond which the CSP will be
unsatisfiable with high probability. Strong refutation is the problem of
certifying that no variable assignment satisfies more than a constant fraction
of clauses; this is the natural algorithmic problem in the unsatisfiable regime
(when ).
Intuitively, strong refutation should become easier as the clause density
grows, because the contradictions introduced by the random clauses become
more locally apparent. For CSPs such as -SAT and -XOR, there is a
long-standing gap between the clause density at which efficient strong
refutation algorithms are known, , and the
clause density at which instances become unsatisfiable with high probability,
.
In this paper, we give spectral and sum-of-squares algorithms for strongly
refuting random -XOR instances with clause density in time or in
rounds of the sum-of-squares hierarchy, for any
and any integer . Our algorithms provide a smooth
transition between the clause density at which polynomial-time algorithms are
known at , and brute-force refutation at the satisfiability
threshold when . We also leverage our -XOR results to obtain
strong refutation algorithms for SAT (or any other Boolean CSP) at similar
clause densities. Our algorithms match the known sum-of-squares lower bounds
due to Grigoriev and Schonebeck, up to logarithmic factors.
Additionally, we extend our techniques to give new results for certifying
upper bounds on the injective tensor norm of random tensors
Erd\H{o}s-Ko-Rado for random hypergraphs: asymptotics and stability
We investigate the asymptotic version of the Erd\H{o}s-Ko-Rado theorem for
the random -uniform hypergraph . For , let and . We show that with probability
tending to 1 as , the largest intersecting subhypergraph of
has size , for any . This lower bound on is
asymptotically best possible for . For this range of and ,
we are able to show stability as well.
A different behavior occurs when . In this case, the lower bound on
is almost optimal. Further, for the small interval , the largest intersecting subhypergraph of
has size , provided that .
Together with previous work of Balogh, Bohman and Mubayi, these results
settle the asymptotic size of the largest intersecting family in
, for essentially all values of and
Short proofs of some extremal results
We prove several results from different areas of extremal combinatorics,
giving complete or partial solutions to a number of open problems. These
results, coming from areas such as extremal graph theory, Ramsey theory and
additive combinatorics, have been collected together because in each case the
relevant proofs are quite short.Comment: 19 page
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