John Lenz

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The Hadwiger number h(G) of a graph G is the maximum size of a complete minor of G. Hadwiger's Conjecture states that h(G) ≥ χ(G). Since χ(G)α(G) ≥ |V (G)|, Hadwiger's Conjecture implies that α(G)h(G) ≥ |V (G)|. We show that (2α(G) − ⌈log τ (τ α(G)/2)⌉)h(G) ≥ |V (G)| where τ ≈ 6.83. For graphs with α(G) ≥ 14, this improves on a recent result of(More)
Let F be a family of r-uniform hypergraphs. The chromatic threshold of F is the infimum of all non-negative reals c such that the subfamily of F comprising hy-pergraphs H with minimum degree at least c |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(More)
is the minimum integer r such that in every edge-coloring of K r by k colors, there is a monochromatic copy of H i in color i for some 1 ≤ i ≤ k. In this paper, we investigate the multicolor Ramsey number r(K 2,t ,. .. , K 2,t , K m), determining the asymptotic behavior up to a polylogarithmic factor for almost all ranges of t and m. Several different(More)
Let k ≥ 2 and F be a linear k-uniform hypergraph with v vertices. We prove that if n is sufficiently large and v|n, then every quasirandom k-uniform hypergraph on n vertices with constant edge density and minimum degree Ω(n k−1) admits a perfect F-packing. The case k = 2 follows immediately from the blowup lemma of Komlós, Sárközy, and Szemerédi. We also(More)
In this paper we consider spectral extremal problems for hypergraphs. We give two general criteria under which such results may be deduced from 'strong stability' forms of the corresponding (pure) extremal results. These results hold for the α-spectral radius defined using the α-norm for any α > 1; the usual spectral radius is the case α = 2. Our results(More)
For each of the notions of hypergraph quasirandomness that have been studied, we identify a large class of hypergraphs F so that every quasirandom hypergraph H admits a perfect F-packing. An informal statement of a special case of our general result for 3-uniform hypergraphs is as follows. Fix an integer r ≥ 4 and 0 < p < 1. Suppose that H is an n-vertex(More)
We show that, for a natural notion of quasirandomness in k-uniform hypergraphs, any quasirandom k-uniform hypergraph on n vertices with constant edge density and minimum vertex degree Ω(n k−1) contains a loose Hamilton cycle. We also give a construction to show that a k-uniform hypergraph satisfying these conditions need not contain a Hamilton-cycle if k −(More)