We prove a 1973 conjecture due to Erdős on the existence of Steiner triple systems with arbitrarily high girth.

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.

Let p ∈ (0, 1/2) be fixed, and let Bn(p) be an n × n random matrix with i.i.d. Bernoulli random variables with mean p. We show that for all t ≥ 0,where sn(Bn(p)) denotes the least singular value of Bn(p) and Cp, ǫp > 0 are constants depending only on p.

Very sparse random graphs are known to typically be singular (i.e., have singular adjacency matrix), due to the presence of "low-degree dependencies" such as isolated vertices and pairs of degree-1 vertices with the same neighbourhood. We prove that these kinds of dependencies are in some sense the only causes of singularity: for constants k ≥ 3 and λ > 0, an Erdős-Rényi random graph G ∼ G(n, λ/n) with n vertices and edge probability λ/n typically has the property that its k-core (its largest subgraph with minimum degree at least k) is nonsingular. This resolves a conjecture of Vu from the 2014 International Congress of Mathematicians, and adds to a short list of known nonsingularity theorems for "extremely sparse" random matrices with density O(1/n). A key aspect of our proof is a technique to extract high-degree vertices and use them to "boost" the rank, starting from approximate rank bounds obtainable from (non-quantitative) spectral convergence machinery due to Bordenave, Lelarge and Salez.

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.

Let ξ be a non-constant real-valued random variable with finite support, and let Mn(ξ) denote an n × n random matrix with entries that are independent copies of ξ. We show that, if ξ is not uniform on its support, then P[Mn(ξ) is singular] = (1 + on(1))P[zero row or column, or two equal (up to sign) rows or columns].For ξ which is uniform on its support, we show that P[Mn(ξ) is singular] = (1 + on(1)) n P[two rows or columns are equal].Corresponding estimates on the least singular value are also provided.

We generalize a map by S. Mason regarding two combinatorial models for key polynomials, in a way that accounts for the major index.We also define similar variants of this map, that regards alternative models for the modified Macdonald polynomials at t = 0, thus partially answer a question by J. Haglund.These maps imply certain uniqueness property regarding inversion-and coinversion-free fillings, which allows us to generalize the notion of charge to a non-symmetric setting, thus answering a question by A. Lascoux. The analogous question in the symmetric setting proves a conjecture by K. Nelson.

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