We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall and Spencer in 2002 [14]. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2−s, (a + 1)2−s] × [b2−t, (b + 1)2−t] for a, b, s, t ∈ ℤ⩾ 0. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n4.09), which implies that the mixing time is at most O(n5.09). We complement this by showing that the relaxation time is at least Ω(n1.38), improving upon the previously best lower bound of Ω(n log n) coming from the diameter of the chain.

Consider a uniform random rooted labelled tree on n vertices. We imagine that each node of the tree has space for a single car to park. A number m ≤ n of cars arrive one by one, each at a node chosen independently and uniformly at random. If a car arrives at a space which is already occupied, it follows the unique path towards the root until it encounters an empty space, in which case it parks there; if there is no empty space, it leaves the tree. Consider m = ⌊α n⌋ and let An,α denote the event that all ⌊α n⌋ cars find spaces in the tree. Lackner and Panholzer proved (via analytic combinatorics methods) that there is a phase transition in this model. Then if α ≤ 1/2, we have $\mathbb{P}({A_{n,\alpha}}) \to {\sqrt{1-2\alpha}}/{(1-\alpha})$, whereas if α > 1/2 we have $\mathbb{P}({A_{n,\alpha}}) \to 0$. We give a probabilistic explanation for this phenomenon, and an alternative proof via the objective method. Along the way, we consider the following variant of the problem: take the tree to be the family tree of a Galton–Watson branching process with Poisson(1) offspring distribution, and let an independent Poisson(α) number of cars arrive at each vertex. Let X be the number of cars which visit the root of the tree. We show that $\mathbb{E}{[X]}$ undergoes a discontinuous phase transition, which turns out to be a generic phenomenon for arbitrary offspring distributions of mean at least 1 for the tree and arbitrary arrival distributions.

We upper-bound the number of common zeros over a finite grid of multivariate polynomials and an arbitrary finite collection of their consecutive Hasse derivatives (in a coordinate-wise sense). To that end, we make use of the tool from Gröbner basis theory known as footprint. Then we establish and prove extensions in this context of a family of well-known results in algebra and combinatorics. These include Alon's combinatorial Nullstellensatz [1], existence and uniqueness of Hermite interpolating polynomials over a grid, estimations of the parameters of evaluation codes with consecutive derivatives [20], and bounds on the number of zeros of a polynomial by DeMillo and Lipton [8], Schwartz [25], Zippel [26, 27] and Alon and Füredi [2]. As an alternative, we also extend the Schwartz-Zippel bound to weighted multiplicities and discuss its connection to our extension of the footprint bound.

In this paper we introduce some Christoffel–Darboux type identities for independence polynomials. As an application, we give a new proof of a theorem of Chudnovsky and Seymour, which states that the independence polynomial of a claw-free graph has only real roots. Another application is related to a conjecture of Merrifield and Simmons.

It follows from known results that every regular tripartite hypergraph of positive degree, with n vertices in each class, has matching number at least n/2. This bound is best possible, and the extremal configuration is unique. Here we prove a stability version of this statement, establishing that every regular tripartite hypergraph with matching number at most (1 + ϵ)n/2 is close in structure to the extremal configuration, where ‘closeness’ is measured by an explicit function of ϵ.

We consider distance colourings in graphs of maximum degree at most d and how excluding one fixed cycle of length ℓ affects the number of colours required as d → ∞. For vertex-colouring and t ⩾ 1, if any two distinct vertices connected by a path of at most t edges are required to be coloured differently, then a reduction by a logarithmic (in d) factor against the trivial bound O(dt) can be obtained by excluding an odd cycle length ℓ ⩾ 3t if t is odd or by excluding an even cycle length ℓ ⩾ 2t + 2. For edge-colouring and t ⩾ 2, if any two distinct edges connected by a path of fewer than t edges are required to be coloured differently, then excluding an even cycle length ℓ ⩾ 2t is sufficient for a logarithmic factor reduction. For t ⩾ 2, neither of the above statements are possible for other parity combinations of ℓ and t. These results can be considered extensions of results due to Johansson (1996) and Mahdian (2000), and are related to open problems of Alon and Mohar (2002) and Kaiser and Kang (2014).

Two graphs G1 and G2 on n vertices are said to pack if there exist injective mappings of their vertex sets into [n] such that the images of their edge sets are disjoint. A longstanding conjecture due to Bollobás and Eldridge and, independently, Catlin, asserts that if (Δ(G1) + 1)(Δ(G2) + 1) ⩽ n + 1, then G1 and G2 pack. We consider the validity of this assertion under the additional assumption that G1 or G2 has bounded codegree. In particular, we prove for all t ⩾ 2 that if G1 contains no copy of the complete bipartite graph K2,t and Δ(G1) > 17t · Δ(G2), then (Δ(G1) + 1)(Δ(G2) + 1) ⩽ n + 1 implies that G1 and G2 pack. We also provide a mild improvement if moreover G2 contains no copy of the complete tripartite graph K1,1,s, s ⩾ 1.

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The classical Kővári–Sós–Turán theorem states that ifGis ann-vertex graph with no copy ofKs,tas a subgraph, then the number of edges inGis at mostO(n2−1/s). We prove that if one forbidsKs,tas aninducedsubgraph, and also forbidsanyfixed graphHas a (not necessarily induced) subgraph, the same asymptotic upper bound still holds, with different constant factors. This introduces a non-trivial angle from which to generalize Turán theory to induced forbidden subgraphs, which this paper explores. Along the way, we derive a non-trivial upper bound on the number of cliques of fixed order in aKr-free graph with no induced copy ofKs,t. This result is an induced analogue of a recent theorem of Alon and Shikhelman and is of independent interest.

This paper is part of the ongoing effort to study high-dimensional permutations. We prove the analogue to the Erdős–Szekeres theorem: For everyk≥ 1, every order-nk-dimensional permutation contains a monotone subsequence of length Ωk($\sqrt{n}$), and this is tight. On the other hand, and unlike the classical case, the longest monotone subsequence in a randomk-dimensional permutation of ordernis asymptotically almost surely Θk(nk/(k+1)).