### An isoperimetric inequality for Hamming balls and local expansion in hypercubes (with Amir Yehudayoff)

Denote by the -dimensional Hamming ball of radius . If and the size of a subset of satisfies , then the vertex boundary of is at least .

Denote by the -dimensional Hamming ball of radius . If and the size of a subset of satisfies , then the vertex boundary of is at least .

Given a system of graphs on the same vertex set , a cooperative coloring for is a choice of vertex sets , such that is independent in and . We give bounds on the minimal such that every graphs with maximum degree have a cooperative coloring, in the cases that (a) the graphs are trees, (b) the graphs are all bipartite.

Any family of sets of edges in an -uniform hypergraph, each having a fractional matchings number of size , has a rainbow matching of size (that is, a set of edges from distinct ’s which supports such a fractional matching).

We complete the line of research on the optimal symmetric rate point in the channel capacity region of the two-user union channel, and we construct a practical near-optimal zero-error communication scheme.

A set of vertices resolves a graph if every vertex is uniquely determined by its vector of distances to the vertices in . The metric dimension of a graph is the minimum cardinality of a resolving set of the graph. We prove that the metric dimension of the Cartesian product of copies of the complete graph on vertices is .

We find the threshold such that, for every , the family of graphs of spectral radius has a finite forbidden subgraphs characterization. As a consequence, we establish the asymptotic formula of the maximum cardinality of equiangular lines with angle in for every .

If the average degree of after deleting any ball of radius is at least , then its second largest eigenvalue is at least .

Journal of Combinatorial Theory, Series B article

A finite family of graphs is -nice if for every graph with (the chromatic number equals to the Ramsey number) and for every coloring of there exists a monochromatic copy of some . It seems conceivable that for any finite family of graphs that contains at least one forest, is -nice for all . We show that the conjecture holds for each of the three families consisting of two connected graphs with 3 edges each.

European Journal of Combinatorics (2018) 72: 29-44 article

Given two disjoint convex polyhedra, we propose a process, based on projections onto the half-spaces defining the two polyhedra, for finding a pair of points, one in each polyhedron, attaining the minimum distance.

Computational Optimization and Applications (2018) 71: 509–523 article

A zone of width on the unit sphere is the set of points within spherical distance of a given great circle. We show that the total width of any collection of zones covering the unit sphere is at least .

Geometric and Functional Analysis (2017) 27: 1367-1377 article

We conjecture that every algebraic hypersurface that gives rise to a -free graph is equivalent, in a suitable sense, to a low-degree hypersurface. We establish a version of this conjecture for -free graphs.

Discrete Mathematics (2018) 341: 1597-1604 article

For each fixed , an -vertex graph not containing a cycle of length has at most edges.

Combinatorics, Probability and Computing (2017) 26: 1-15 article erratum

Let be disjoint -point sets in . A colorful simplex is the convex hull of points each of which comes from a distinct . There always exists a point that is contained in at least colorful simplices.

The Electronic Journal of Combinatorics (2014) 21: P4.39 article