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Let $\mathcal{F}$ be a family of r-uniform hypergraphs. The chromatic threshold of $\mathcal{F}$ is the infimum of all non-negative reals c such that the subfamily of $\mathcal{F}$ comprising hypergraphs H with minimum degree at least $c \binom{| V(H) |}{r-1}$ has bounded chromatic number. This parameter has a long history for graphs (r = 2), and in this paper we begin its systematic study for hypergraphs.
Łuczak and Thomassé recently proved that the chromatic threshold of the so-called near bipartite graphs is zero, and our main contribution is to generalize this result to r-uniform hypergraphs. For this class of hypergraphs, we also show that the exact Turán number is achieved uniquely by the complete (r + 1)-partite hypergraph with nearly equal part sizes. This is one of very few infinite families of non-degenerate hypergraphs whose Turán number is determined exactly. In an attempt to generalize Thomassen's result that the chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the chromatic threshold of the family of 3-uniform hypergraphs not containing {abc, abd, cde}, the so-called generalized triangle.
In order to prove upper bounds we introduce the concept of fibre bundles, which can be thought of as a hypergraph analogue of directed graphs. This leads to the notion of fibre bundle dimension, a structural property of fibre bundles that is based on the idea of Vapnik–Chervonenkis dimension in hypergraphs. Our lower bounds follow from explicit constructions, many of which use a hypergraph analogue of the Kneser graph. Using methods from extremal set theory, we prove that these Kneser hypergraphs have unbounded chromatic number. This generalizes a result of Szemerédi for graphs and might be of independent interest. Many open problems remain.
This collection of essays pays tribute to Nancy Freeman Regalado, a ground-breaking scholar in the field of medieval French literature whose research has always pushed beyond disciplinary boundaries. The articles in the volume reflect the depth and diversity of her scholarship, as well as her collaborations with literary critics, philologists, historians, art historians, musicologists, and vocalists - in France, England, and the United States. Inspired by her most recent work, these twenty-four essays are tied together by a single question, rich in ramifications: how does performance shape our understanding of medieval and pre-modern literature and culture, whether the nature of that performance is visual, linguistic, theatrical, musical, religious, didactic, socio-political, or editorial? The studies presented here invite us to look afresh at the interrelationship of audience, author, text, and artifact, to imagine new ways of conceptualizing the creation, transmission, and reception of medieval literature, music, and art.
EGLAL DOSS-QUINBY is Professor of French at Smith College; ROBERTA L. KRUEGER is Professor of French at Hamilton College; E. JANE BURNS is Professor of Women's Studies and Adjunct Professor of Comparative Literature at the University of North Carolina, Chapel Hill.
Contributors: ANNE AZÉMA, RENATE BLUMENFELD-KOSINSKI, CYNTHIA J. BROWN, ELIZABETH A. R. BROWN, MATILDA TOMARYN BRUCKNER, E. JANE BURNS, ARDIS BUTTERFIELD, KIMBERLEE CAMPBELL, ROBERT L. A. CLARK, MARK CRUSE, KATHRYN A. DUYS, ELIZABETH EMERY, SYLVIA HUOT, MARILYN LAWRENCE, KATHLEEN A. LOYSEN, LAURIE POSTLEWATE, EDWARD H. ROESNER, SAMUEL N. ROSENBERG, LUCY FREEMAN SANDLER, PAMELA SHEINGORN, HELEN SOLTERER, JANE H. M. TAYLOR, EVELYN BIRGE VITZ, LORI J. WALTERS, AND MICHEL ZINK.