Boris Bukh

Sums of dilates

The lambda-dilate of a set A is lambda*A={lambda a : a \in A}. We give an asymptotically sharp lower bound on the size of sumsets of the form lambda_1*A+...+lambda_k*A for arbitrary integers lambda_1,...,lambda_k and integer sets A. We also establish an upper bound for such sums, which is similar to, but often stronger than Plunneckes inequality.

A bound on the number of edges in graphs without an even cycle

We show that, for each fixed k , an n -vertex graph not containing a cycle of length 2 k has at most

Sum–product estimates for rational functions

80\sqrt{k\log k}\cdot n^{1+1/k}+O(n)

Turán numbers for K s,t -free graphs: Topological obstructions and algebraic constructions

edges.

Bounds on equiangular lines and on related spherical codes

We establish several sum-product estimates over finite fields that involve polynomials and rational functions. First, |f(A)+f(A)|+|AA| is substantially larger than |A| for an arbitrary polynomial f over F_p. Second, a characterization is given for the rational functions f and g for which |f(A)+f(A)|+|g(A,A)| can be as small as |A|, for large |A|. Third, we show that under mild conditions on f, |f(A,A)| is substantially larger than |A|, provided |A| is large. We also present a conjecture on what the general sum-product result should be.

Induced subgraphs of Ramsey graphs with many distinct degrees

We show that every hypersurface in ℝs × ℝs contains a large grid, i.e., the set of the form S × T, with S, T ⊂ ℝs. We use this to deduce that the known constructions of extremal K2,2-free and K3,3-free graphs cannot be generalized to a similar construction of Ks,s-free graphs for any s ≥ 4. We also give new constructions of extremal Ks,t-free graphs for large t.

Non-trivial solutions to a linear equation in integers

An

An Improvement of the Beck–Fiala Theorem

L

Longest Common Subsequences in Sets of Words

-spherical code is a set of Euclidean unit vectors whose pairwise inner products belong to the set

One-Sided Epsilon-Approximants

L