Human infants develop a variety of attentional mechanisms that allow them to extract relevant information from a cluttered multimodal world. We know that both social and nonsocial cues shift infants' attention, but not how these cues differentially affect learning of multimodal events. Experiment 1 used social cues to direct 8- and 4-month-olds' attention to two audiovisual events (i.e., animat...

In 1982 (see [1]) Imre Bárány observed that some of the classical theorems in convexity admit interesting and mysterious generalizations which he called “colorful theorems”. For example, the Colorful Helly Theorem says that if a family (repetitions of the same sets are allowed) of compact convex sets in Rk is colored (properly) with k+1 colors and it has the property that any choice of k+1 diff...

An m-colored digraph is a digraph whose arcs are colored with m colors. A directed path is monochromatic when its arcs are colored alike. A set S ⊆ V (D) is a kernel by monochromatic paths whenever the two following conditions hold: 1. For any x, y ∈ S, x 6= y, there is no monochromatic directed path between them. 2. For each z ∈ (V (D)− S) there exists a zS-monochromatic directed path. In this...

We call the digraphD anm-coloured digraph if its arcs are coloured withm colours. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike. Let D be an m-coloured digraph. A set N ⊆ V (D) is said to be a kernel by monochromatic paths if it satisfies the following two conditions: (i) for every pair of different vertices u, v ∈ N there is no monochromati...

We demonstrate that the topological Helly theorem and the algebraic Auslander-Buchsbaum may be viewed as different versions of the same phenomenon. Using this correspondence we show how the colorful Helly theorem of I.Barany and its generalizations by G.Kalai and R.Meshulam translates to the algebraic side. Our main results are algebraic generalizations of these translations, which in particula...

We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike. A set N ⊆ V (D) is said to be a kernel by monochromatic paths if it satisfies the two following conditions (i) for every pair of different vertices u, v ∈ N there is no monochromatic directed path between t...

Fundamental n-groupoids for a topological space are introduced, by techniques based on Moore paths, similar to those used in [G3] for symmetric simplicial sets. Also the 'directed case' is treated, based on a structure introduced in [G6]: a directed topological space, where privileged directions are assigned and paths need not be reversible; such objects are provided with fundamental n-categori...

An asteroidal triple is a stable set of three vertices such that each pair is connected by a path avoiding the neighborhood of the third vertex. Asteroidal triples play a central role in a classical characterization of interval graphs by Lekkerkerker and Boland. Their result says that a chordal graph is an interval graph if and only if it contains no asteroidal triple. In this paper, we prove a...

Fundamental n-groupoids for a topological space are introduced, by techniques based on Moore paths, similar to those used in [G3] for symmetric simplicial sets. Also the 'directed case' is treated, based on a structure introduced in [G6]: a directed topological space, where privileged directions are assigned and paths need not be reversible; such objects are provided with fundamental n-categori...