We treat problems of fair division, their various interconnections, and their relations to Sperner's lemma and the KKM theorem as well as their variants. We prove extensions of Alon's necklace splitting result in certain regimes and relate it to hyperplane mass partitions.We show the existence of fair cake division and rental harmony in the sense of Su even in the absence of full information. Furthermore, we extend Sperner's lemma and the KKM theorem to (colorful) quantitative versions for polytopes and pseudomanifolds. For simplicial polytopes our results turn out to be improvements over the earlier work of De Loera, Peterson, and Su on a polytopal version of Sperner's lemma. Moreover, our results extend the work of Musin on quantitative Sperner-type results for PL manifolds.

Settling Kahn's conjecture (2001), we prove the following upper bound on the number i(G) of independent sets in a graph G without isolated vertices:where du is the degree of vertex u in G. Equality occurs when G is a disjoint union of complete bipartite graphs. The inequality was previously proved for regular graphs by Kahn and Zhao.We also prove an analogous tight lower bound:where equality occurs for G a disjoint union of cliques. More generally, we prove bounds on the weighted versions of these quantities, i.e., the independent set polynomial, or equivalently the partition function of the hard-core model with a given fugacity on a graph.

AbstractMay the triforce be the 3-uniform hypergraph on six vertices with edges {123′, 12′3, 1′23}. We show that the minimum triforce density in a 3-uniform hypergraph of edge density δ is δ4–o(1) but not O(δ4).Let M(δ) be the maximum number such that the following holds: for every ∊ > 0 and $G = {\mathbb{F}}_2^n$ with n sufficiently large, if A ⊆ G × G with A ≥ δ|G|2, then there exists a nonzero “popular difference” d ∈ G such that the number of “corners” (x, y), (x + d, y), (x, y + d) ∈ A is at least (M(δ)–∊)|G|2. As a corollary via a recent result of Mandache, we conclude that M(δ) = δ4–o(1) and M(δ) = ω(δ4).On the other hand, for 0 < δ < 1/2 and sufficiently large N, there exists A ⊆ [N]3 with |A| ≥ δN3 such that for every d ≠ 0, the number of corners (x, y, z), (x + d, y, z), (x, y + d, z), (x, y, z + d) ∈ A is at most δc log(1/δ)N3. A similar bound holds in higher dimensions, or for any configuration with at least 5 points or affine dimension at least 3.

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Let H be a graph allowing loops as well as vertex-and edge-weights. We prove that, for every triangle-free graph G without isolated vertices, the weighted number of graph homomorphisms hom(G, H) satisfies the inequalitywhere du denotes the degree of vertex u in G. In particular, one has hom(G, H) 1/|E(G)| ≤ hom (K d,d , H) 1/d 2 for every d-regular triangle-free G. The triangle-free hypothesis on G is best possible. More generally, we prove a graphical Brascamp-Lieb type inequality, where every edge of G is assigned some two-variable function. These inequalities imply tight upper bounds on the partition function of various statistical models such as the Ising and Potts models, which includes independent sets and graph colorings.For graph colorings, corresponding to H = Kq (also valid if some of the vertices of Kq are looped), we show that the triangle-free hypothesis on G may be dropped. A corollary is that among d-regular graphs, G = K d,d maximizes the quantity cq(G) 1/|V (G)| for every q and d, where cq(G) counts proper q-colorings of G.Finally, we show that if the edge-weight matrix of H is positive semidefinite, thenThis implies that among d-regular graphs, G = K d+1 maximizes hom(G, H) 1/|V (G)| . For 2-spin Ising models, our results give a complete characterization of extremal graphs: complete bipartite graphs maximize the partition function of 2-spin antiferromagnetic models and cliques maximize the partition function of ferromagnetic models. These results settle a number of conjectures by Galvin-Tetali, Galvin, and Cohen-Csikvári-Perkins-Tetali, and provide an alternate proof to a conjecture by Kahn. arXiv:1809.09462v2 [math.CO] , where denotes a disjoint union. Question 1.1 was initially raised by Granville in 1988 in connection with the Cameron-Erdős conjecture on the number of sum-free sets. Alon [1] and Kahn [27] conjectured that G = K d,d is the exact maximizer. Alon [1] proved an asymptotic version as d → ∞, Kahn [27] proved the exact version under the additional hypothesis that G is bipartite, and Zhao [38] later removed this bipartite assumption. The results of Kahn [27] and Zhao [38] together answer Question 1.1: the maximizer is K d,d (unique up to taking disjoint unions of copies of itself). Galvin and Tetali [22] initiated the study of Questions 1.2 and 1.3 and extended Kahn's entropy method [27] to prove that, under the additional hypothesis that G is bipartite, G = K d,d is also the maximizer for hom(G, H) 1/|V (G)| . See Lubetzky and Zhao [32, Section 6] for a different proof using Hölder/Brascamp-Lieb type inequalities. Can the bipartite hypothesis on G also be dropped in this case? Not for all H: e.g., for H = , G = K d+1 is the maximizer instead of K d,d . Extending the technique for independent sets, Zhao [39] showed that the bipartite hypothesis can be dropped for certain classes of H, but the techniques failed for H = K q , corresponding to colorings (Question 1.2). It remained a tantalizing conjecture to remove the bipartite hypothesis for colorings.Recently, Davies, Jenssen, Perki...