For x real, let {x} be the fractional part of x (i.e. {x} = x − bxc). In this paper we prove the k = 5 case of the following conjecture (the lonely runner conjecture): for any k positive reals v1, . . . , vk there exists a real number t such that 1/(k + 1) ≤ {vit} ≤ k/(k + 1) for i = 1, . . . , k.

We consider the problem whether a permutation of a finite set is uniquely determined by its identification minors. While there exist non-reconstructible permutations of every set with two, three, or four elements, we show that every permutation of a finite set with at least five elements is reconstructible from its identification minors. Moreover, we provide an algorithm for recovering a permut...

Motivated by an old problem known as Ryser’s Conjecture, we prove that for r = 4 and r = 5, there exists > 0 such that every r-partite r-uniform hypergraph H has a cover of size at most (r − )ν(H), where ν(H) denotes the size of a largest matching in H.

Tutte proved that every 3-connected graph on more than 4 nodes has a contractible edge. Barnette and Grünbaum proved the existence of a removable edge in the same setting. We show that the sequence of contractions and the sequence of removals from G to the K4 can be computed in O(|V | ) time by extending Barnette and Grünbaum’s theorem. As an application, we derive a certificate for the 3-conne...

We prove that for d ≥ 3, the 1-skeleton of any (d− 1)-dimensional doubly Cohen-Macaulay (abbreviated 2-CM) complex is generically drigid. This implies that Barnette’s lower bound inequalities for boundary complexes of simplicial polytopes ([4],[3]) hold for every 2-CM complex of dimension ≥ 2 (see Kalai [8]). Moreover, the initial part (g0, g1, g2) of the g-vector of a 2-CM complex (of dimensio...

Suppose σ ∈ Lω1,ω(L) is such that all equations occurring in σ are positive, have the same set of variables on each side of the equality symbol, and have at least one function symbol on each side of the equality symbol. We show that σ satisfies Vaught’s conjecture. In particular this proves Vaught’s conjecture for sentences of Lω1,ω(L) without equality.

The chromatic number of the categorical product of two n-tournaments can be strictly smaller than n. We show that min{χ(S × T ) : S and T are n-tournaments} is asymptotically equal to λn, where 12 ≤ λ ≤ 2 3 .

We develop a theory for the Hermite-Krichever Ansatz on the Heun equation. As a byproduct, we find formulae which reduce hyperelliptic integrals to elliptic ones.