1,135 research outputs found
Sums and Products with Smooth Numbers
We estimate the sizes of the sumset A + A and the productset A A in
the special case that A = S (x, y), the set of positive integers n less than or
equal to x, free of prime factors exceeding y.Comment: 12 page
Upper bounds for sunflower-free sets
A collection of sets is said to form a -sunflower, or -system,
if the intersection of any two sets from the collection is the same, and we
call a family of sets sunflower-free if it contains no
sunflowers. Following the recent breakthrough of Ellenberg and Gijswijt and
Croot, Lev and Pach we apply the polynomial method directly to
Erd\H{o}s-Szemer\'{e}di sunflower problem and prove that any sunflower-free
family of subsets of has size at most We say that
a set for is
sunflower-free if every distinct triple there exists a coordinate
where exactly two of are equal. Using a version of the
polynomial method with characters
instead of polynomials, we
show that any sunflower-free set has size
where . This can be
seen as making further progress on a possible approach to proving the
Erd\H{o}s-Rado sunflower conjecture, which by the work of Alon, Sphilka and
Umans is equivalent to proving that for some constant
independent of .Comment: 5 page
Hardness of Exact Distance Queries in Sparse Graphs Through Hub Labeling
A distance labeling scheme is an assignment of bit-labels to the vertices of
an undirected, unweighted graph such that the distance between any pair of
vertices can be decoded solely from their labels. An important class of
distance labeling schemes is that of hub labelings, where a node
stores its distance to the so-called hubs , chosen so that for
any there is belonging to some shortest
path. Notice that for most existing graph classes, the best distance labelling
constructions existing use at some point a hub labeling scheme at least as a
key building block. Our interest lies in hub labelings of sparse graphs, i.e.,
those with , for which we show a lowerbound of
for the average size of the hubsets.
Additionally, we show a hub-labeling construction for sparse graphs of average
size for some , where is the
so-called Ruzsa-Szemer{\'e}di function, linked to structure of induced
matchings in dense graphs. This implies that further improving the lower bound
on hub labeling size to would require a
breakthrough in the study of lower bounds on , which have resisted
substantial improvement in the last 70 years. For general distance labeling of
sparse graphs, we show a lowerbound of , where is the communication complexity of the
Sum-Index problem over . Our results suggest that the best achievable
hub-label size and distance-label size in sparse graphs may be
for some
- …