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Palettes determine uniform Tur\'an density
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abstract
Tur\'an problems, which concern the minimum density threshold required for the existence of a particular substructure, are among the most fundamental problems in extremal combinatorics. We study Tur\'an problems for hypergraphs with an additional uniformity condition on the edge distribution. This kind of Tur\'an problems was introduced by Erd\H{o}s and S\'os in the 1980s but it took more than 30 years until the first non-trivial exact results were obtained when Glebov, Kr\'al' and Volec [Israel J. Math. 211 (2016), 349--366] and Reiher, R\"odl and Schacht [J. Eur. Math. Soc. 20 (2018), 1139--1159] determined the uniform Tur\'an density of $K_4^{(3)-}$. Subsequent results exploited the powerful hypergraph regularity method, developed by Gowers and by Nagle, R\"odl and Schacht about two decades ago. Central to the study of the uniform Tur\'an density of hypergraphs are palette constructions, which were implicitly introduced by R\"odl in the 1980s. We prove that palette constructions always yield tight lower bounds, unconditionally confirming present empirical evidence. This results in new and simpler approaches to determining uniform Tur\'an densities, which completely bypass the use of the hypergraph regularity method.
Forward citations
Cited by 6 Pith papers
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For 3-uniform hypergraphs, every density in an interval ending at 1 is exactly the uniform Turán density of some possibly infinite forbidden family.
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Lagrangians are attained as uniform Tur\'an densities
Every scaled Lagrangian of a 3-graph is realized as a uniform Turán density, and the set of uniform Turán densities has accumulation points.
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Finite palette endpoints and degree-square Tur\'an problems
Proves exact degree-square Turán formulas for tournament palettes via auxiliary digraphs and majorization, yielding finite 3-graphs with uniform densities approaching 1/3.
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Property O and Erd\H{o}s--Szekeres properties in linear hypergraphs
The minimum number of edges in a linear oriented k-graph with Property O is bracketed between (k!)²/(2e²k⁴) and (1+o(1))·4k⁶ ln²k·(k!)², settling it up to a polynomial factor.
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Uniform Tur\'an density beyond 3-graphs
For every r≥3 there exist r-uniform hypergraphs whose uniform Turán density is 1/4, and others whose uniform Turán density is binom(r,2)^{-binom(r,2)}; the r≥5 cases are the first explicit non-zero values.
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Uniform Tur\'an density -- palette classification
A hypergraph exists that is colorable by each of given palettes but not by an additional palette exactly when no palette homomorphism exists to that additional palette or its inverse.
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