Oleg Pikhurko

Minimum number of k-Cliques in graphs with bounded independence number

Erdős asked in 1962 about the value of f(n,k,l), the minimum number of k-cliques in a graph with order n and independence number less than l. The case (k,l)=(3,3) was solved by Lorden. Here we solve the problem (for all large n) for (3,l) with 4 ≤ l ≤ 7 and (k,3) with 4 ≤ k ≤ 7. Independently, Das, Huang, Ma, Naves and Sudakov resolved the cases (k,l)=(3,4) and (4,3).

Lattice points in lattice polytopes

It is shown that, for any lattice polytope P ⊂ℝ d the set int ( P )∩lℤ d (provided that it is non-empty) contains a point whose coefficient of asymmetry with respect to P is at most 8 d · (8 l +7) 2 2 d +1 . If, moreover, P is a simplex, then this bound can be improved to 8 · (8 l +7 ) 2 d +1 . As an application, new upper bounds on the volume of a lattice polytope are deduced, given its dimension and the number of sublattice points in its interior.

On Triple Systems with Independent Neighbourhoods

Let

An exact Turán result for the generalized triangle

{\cal H}

Asymptotic Structure of Graphs with the Minimum Number of Triangles

be a 3-uniform hypergraph on an

Maximizing the number of q-colorings

n

The maximum number of K3-free and K4-free edge 4-colorings

-element vertex set

An analytic approach to stability

V

A note on the Turán function of even cycles

. The neighbourhood of

Bounds on the Generalised Acyclic Chromatic Numbers of Bounded Degree Graphs

a,b\in V