A constructive proof of Ky Fan’s coincidence theorem

A constructive proof of Ky Fan's generalization of Tucker's lemma

We present a constructive proof of Ky Fan’s combinatorial lemma concerning labellings of triangulated spheres. Our construction works for triangulations of Sn that contain a flag of hemispheres. As a consequence, we produce a constructive proof of Tucker’s lemma that holds for a larger class of triangulations than previous constructive proofs.

A Constructive Proof of Gleason's Theorem

A Constructive Proof of Vizing's Theorem

We consider finite graphs with no self-loops and no multiple edges. A graph is valid if all edges incident on a vertex have different colors. We prove Vizing’s Theorem. All the edges of a graph of maximum degree less than N can be colored using N colors so that the graph is valid. We call a color incident on a vertex if an edge incident on that vertex has that color; otherwise, the color is fre...

A constructive proof of Orzech’s theorem

Corollary 0.2 is a well-known fact (e.g., it appears in [12, Lemma A.3] and in [3]), but most of its proofs in literature do not generalize to Theorem 0.1. Orzech’s original proof of Theorem 0.1 (with the corrections provided in [2], as the original version was shaky) proceeds by reducing the theorem to the case when A is Noetherian, and then using this Noetherianness in an elegant and yet myst...

Constructive Proof of the Carpenter’s Theorem

We give a constructive proof of Carpenter’s Theorem due to Kadison [14, 15]. Unlike the original proof our approach also yields the real case of this theorem. 1. Kadison’s theorem In [14] and [15] Kadison gave a complete characterization of the diagonals of orthogonal projections on a Hilbert space H. Theorem 1.1 (Kadison). Let {di}i∈I be a sequence in [0, 1]. Define a = ∑ di<1/2 di and b = ∑ d...