F-free hypergraphs have bounded strong chromatic number except for the 3-uniform star expansion S_k^+, where bounds are asymptotically sharp as k to infinity; similar characterizations and sharp bounds hold when forbidding Berge copies of graphs F.
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For large n, any n-vertex r-uniform hypergraph with matching number < s has spectral radius at most that of F_{s-1}(n), with equality only for that hypergraph.
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On the largest chromatic number of $F$-free hypergraphs
F-free hypergraphs have bounded strong chromatic number except for the 3-uniform star expansion S_k^+, where bounds are asymptotically sharp as k to infinity; similar characterizations and sharp bounds hold when forbidding Berge copies of graphs F.
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A Spectral Confirmation of the Erd\H{o}s Matching Conjecture
For large n, any n-vertex r-uniform hypergraph with matching number < s has spectral radius at most that of F_{s-1}(n), with equality only for that hypergraph.