Maria Axenovich

Classical and Quantum Magnetism in Giant Keplerate Magnetic Molecules

Complementary theoretical modeling methods are presented for the classical and quantum Heisenberg model to explain the magnetic properties of nanometer-sized magnetic molecules. Excellent quantitative agreement is achieved between our experimental data down to 0.1 K and for fields up to 60 Tesla and our theoretical results for the giant Keplerate species {Mo72Fe30}, by far the largest paramagnetic molecule synthesized to date.

Q 2 -free Families in the Boolean Lattice

For a family

On multiple coverings of the infinite rectangular grid with balls of constant radius

{{\cal F}}

On a Generalized Anti-Ramsey Problem

of subsets of [n] = {1, 2, ..., n} ordered by inclusion, and a partially ordered set P, we say that

Local Anti-Ramsey Numbers of Graphs

{{\cal F}}

A generalized Ramsey problem

is P-free if it does not contain a subposet isomorphic to P. Let ex(n, P) be the largest size of a P-free family of subsets of [n]. Let Q2 be the poset with distinct elements a, b, c, d, a < b,c < d; i.e., the 2-dimensional Boolean lattice. We show that 2N − o(N) ≤ ex(n, Q2) ≤ 2.283261N + o(N), where

Bipartite anti-Ramsey numbers of cycles: BIPARTITE ANTI-RAMSEY NUMBERS

N = \binom{n}{\lfloor n/2 \rfloor}

A regularity lemma and twins in words

. We also prove that the largest Q2-free family of subsets of [n] having at most three different sizes has at most 2.20711N members.

WHEN DO THREE LONGEST PATHS HAVE A COMMON VERTEX

We consider the coverings of graphs with balls of constant radius satisfying special multiplicity condition. A (t,i,j)-cover of a graph G=(V,E) is a subset S of vertices, such that every element of S belongs to exactly i balls of radius t centered at elements of S and every element of V@?S belongs to exactly j balls of radius t centered at elements of S. For the infinite rectangular grid, we show that in any (t,i,j)-cover, i and j differ by at most t+2 except for one degenerate case. Furthermore, for i and j satisfying |i-j|>4 we show that all (t,i,j)-covers are the unions of the diagonals periodically located in the grid. Also, we give the description of all (1,i,j)-covers.

Sub-Ramsey Numbers for Arithmetic Progressions

For positive integers , a coloring of is called a -coloring if the edges of every receive at least and at most colors. Let denote the maximum number of colors in a -coloring of . Given we determine the largest such that all -colorings of have at most O(n) colors and we determine asymptotically when it is of order equal to . We give several bounds and constructions.