The hedgehog Ht is a 3-uniform hypergraph on vertices $1, \ldots ,t + \left({\matrix{t \cr 2}}\right)$ such that, for any pair (i, j) with 1 ≤ i < j ≤ t, there exists a unique vertex k > t such that {i, j, k} is an edge. Conlon, Fox and Rödl proved that the two-colour Ramsey number of the hedgehog grows polynomially in the number of its vertices, while the four-colour Ramsey number grows exponentially in the square root of the number of vertices. They asked whether the two-colour Ramsey number of the hedgehog Ht is nearly linear in the number of its vertices. We answer this question affirmatively, proving that r(Ht) = O(t2 ln t).

For $$\tau \in {S_3}$$, let $$\mu _n^\tau $$ denote the uniformly random probability measure on the set of $$\tau $$-avoiding permutations in $${S_n}$$. Let $${\mathbb {N}^*} = {\mathbb {N}} \cup \{ \infty \} $$ with an appropriate metric and denote by $$S({\mathbb{N}},{\mathbb{N}^*})$$ the compact metric space consisting of functions $$\sigma {\rm{= }}\{ {\sigma _i}\} _{i = 1}^\infty {\rm{}}$$ from $$\mathbb {N}$$ to $${\mathbb {N}^ * }$$ which are injections when restricted to $${\sigma ^{ - 1}}(\mathbb {N})$$; that is, if $${\sigma _i}{\rm{= }}{\sigma _j}$$, $$i \ne j$$, then $${\sigma _i} = \infty $$. Extending permutations $$\sigma \in {S_n}$$ by defining $${\sigma _j} = j$$, for $$j \gt n$$, we have $${S_n} \subset S({\mathbb{N}},{{\mathbb{N}}^*})$$. For each $$\tau \in {S_3}$$, we study the limiting behaviour of the measures $$\{ \mu _n^\tau \} _{n = 1}^\infty $$ on $$S({\mathbb{N}},{\mathbb{N}^*})$$. We obtain partial results for the permutation $$\tau= 321$$ and complete results for the other five permutations $$\tau \in {S_3}$$.

Motivated by the work of Razborov about the minimal density of triangles in graphs we study the minimal density of the 5-cycle C5. We show that every graph of order n and size $ (1 - 1/k) \left( {\matrix{n \cr 2 }} \right) $, where k ≥ 3 is an integer, contains at least

$$({1 \over {10}} - {1 \over {2k}} + {1 \over {{k^2}}} - {1 \over {{k^3}}} + {2 \over {5{k^4}}}){n^5} + o({n^5})$$

The famous Erdős–Gallai theorem on the Turán number of paths states that every graph with n vertices and m edges contains a path with at least (2m)/n edges. In this note, we first establish a simple but novel extension of the Erdős–Gallai theorem by proving that every graph G contains a path with at least

$${{(s + 1){N_{s + 1}}(G)} \over {{N_s}(G)}} + s - 1$$

Given a graph G and a bijection f : E(G) → {1, 2,…,e(G)}, we say that a trail/path in G is f-increasing if the labels of consecutive edges of this trail/path form an increasing sequence. More than 40 years ago Chvátal and Komlós raised the question of providing worst-case estimates of the length of the longest increasing trail/path over all edge orderings of Kn. The case of a trail was resolved by Graham and Kleitman, who proved that the answer is n-1, and the case of a path is still wide open. Recently Lavrov and Loh proposed studying the average-case version of this problem, in which the edge ordering is chosen uniformly at random. They conjectured (and Martinsson later proved) that such an ordering with high probability (w.h.p.) contains an increasing Hamilton path.

In this paper we consider the random graph G = Gn,p with an edge ordering chosen uniformly at random. In this setting we determine w.h.p. the asymptotics of the number of edges in the longest increasing trail. In particular we prove an average-case version of the result of Graham and Kleitman, showing that the random edge ordering of Kn has w.h.p. an increasing trail of length (1-o(1))en, and that this is tight. We also obtain an asymptotically tight result for the length of the longest increasing path for random Erdős-Renyi graphs with p = o(1).

This paper explores the possible use of Schubert cells and Schubert varieties in finite geometry, particularly in regard to the question of whether these objects might be a source of understanding of ovoids or provide new examples. The main result provides a characterization of those Schubert cells for finite Chevalley groups which have the first property (thinness) of ovoids. More importantly, perhaps this short paper can help to bridge the modern language barrier between finite geometry and representation theory. For this purpose, this paper includes very brief surveys of the powerful lattice theory point of view from finite geometry and the powerful method of indexing points of flag varieties by Chevalley generators from representation theory.

The induced removal lemma of Alon, Fischer, Krivelevich and Szegedy states that if an n-vertex graph G is ε-far from being induced H-free then G contains δH(ε) · nh induced copies of H. Improving upon the original proof, Conlon and Fox proved that 1/δH(ε)is at most a tower of height poly(1/ε), and asked if this bound can be further improved to a tower of height log(1/ε). In this paper we obtain such a bound for graphs G of density O(ε). We actually prove a more general result, which, as a special case, also gives a new proof of Fox’s bound for the (non-induced) removal lemma.

We prove that the free uniform spanning forest of any bounded degree proper plane graph is connected almost surely, answering a question of Benjamini, Lyons, Peres and Schramm. We provide a quantitative form of this result, calculating the critical exponents governing the geometry of the uniform spanning forests of transient proper plane graphs with bounded degrees and codegrees. We find that the same exponents hold universally over this entire class of graphs provided that measurements are made using the hyperbolic geometry of their circle packings rather than their usual combinatorial geometry.

In this work, we use rigorous probabilistic methods to study the asymptotic degree distribution, clustering coefficient, and diameter of geographical attachment networks. As a type of small-world network model, these networks were first proposed in the physical literature, where they were analyzed only with heuristic arguments and computational simulations.

In an earlier paper, the present authors (2015) introduced the altermatic number of graphs and used Tucker’s lemma, an equivalent combinatorial version of the Borsuk–Ulam theorem, to prove that the altermatic number is a lower bound for chromatic number. A matching Kneser graph is a graph whose vertex set consists of all matchings of a specified size in a host graph and two vertices are adjacent if their corresponding matchings are edge-disjoint. Some well-known families of graphs such as Kneser graphs, Schrijver graphs and permutation graphs can be represented by matching Kneser graphs. In this paper, unifying and generalizing some earlier works by Lovász (1978) and Schrijver (1978), we determine the chromatic number of a large family of matching Kneser graphs by specifying their altermatic number. In particular, we determine the chromatic number of these matching Kneser graphs in terms of the generalized Turán number of matchings.

We introduce two new bases of the ring of polynomials and study their relations to known bases. The first basis is the quasi-Lascoux basis, which is simultaneously both a $K$-theoretic deformation of the quasi-key basis and also a lift of the $K$-analogue of the quasi-Schur basis from quasi-symmetric polynomials to general polynomials. We give positive expansions of this quasi-Lascoux basis into the glide and Lascoux atom bases, as well as a positive expansion of the Lascoux basis into the quasi-Lascoux basis. As a special case, these expansions give the first proof that the $K$-analogues of quasi-Schur polynomials expand positively in multifundamental quasi-symmetric polynomials of T. Lam and P. Pylyavskyy.

The second new basis is the kaon basis, a $K$-theoretic deformation of the fundamental particle basis. We give positive expansions of the glide and Lascoux atom bases into this kaon basis.

Throughout, we explore how the relationships among these $K$-analogues mirror the relationships among their cohomological counterparts. We make several “alternating sum” conjectures that are suggestive of Euler characteristic calculations.

The no restart random walk (NRRW) is a random network growth model driven by a random walk that builds the graph while moving on it, adding and connecting a new leaf node to the current position of the walker every s steps. We show a fundamental dichotomy in NRRW with respect to the parity of s: for ${s}=1$ we prove that the random walk is transient and non-leaf nodes have degrees bounded above by an exponential distribution; for s even we prove that the random walk is recurrent and non-leaf nodes have degrees bounded below by a power law distribution. These theoretical findings highlight and confirm the diverse and rich behaviour of NRRW observed empirically.

We study the average nearest-neighbour degree a(k) of vertices with degree k. In many real-world networks with power-law degree distribution, a(k) falls off with k, a property ascribed to the constraint that any two vertices are connected by at most one edge. We show that a(k) indeed decays with k in three simple random graph models with power-law degrees: the erased configuration model, the rank-1 inhomogeneous random graph, and the hyperbolic random graph. We find that in the large-network limit for all three null models, a(k) starts to decay beyond $n^{(\tau-2)/(\tau-1)}$ and then settles on a power law $a(k)\sim k^{\tau-3}$, with $\tau$ the degree exponent.

We consider subgraph counts in general preferential attachment models with power-law degree exponent $\tau > 2$. For all subgraphs H, we find the scaling of the expected number of subgraphs as a power of the number of vertices. We prove our results on the expected number of subgraphs by defining an optimization problem that finds the optimal subgraph structure in terms of the indices of the vertices that together span it and by using the representation of the preferential attachment model as a Pólya urn model.

Archdeacon and Grable (1995) proved that the genus of the random graph $G\in {\mathcal{G}}_{n,p}$ is almost surely close to $pn^{2}/12$ if $p=p(n)\geqslant 3(\ln n)^{2}n^{-1/2}$. In this paper we prove an analogous result for random bipartite graphs in ${\mathcal{G}}_{n_{1},n_{2},p}$. If $n_{1}\geqslant n_{2}\gg 1$, phase transitions occur for every positive integer $i$ when $p=\unicode[STIX]{x1D6E9}((n_{1}n_{2})^{-i/(2i+1)})$. A different behaviour is exhibited when one of the bipartite parts has constant size, i.e., $n_{1}\gg 1$ and $n_{2}$ is a constant. In that case, phase transitions occur when $p=\unicode[STIX]{x1D6E9}(n_{1}^{-1/2})$ and when $p=\unicode[STIX]{x1D6E9}(n_{1}^{-1/3})$.

Let $p>3$ be a prime and let $a$ be a rational $p$-adic integer with $a\not \equiv 0\;(\text{mod}\;p)$. We evaluate

$$\begin{eqnarray}\mathop{\sum }_{k=1}^{(p-1)/2}\frac{1}{k}\binom{a}{k}\binom{-1-a}{k}\quad \text{and}\quad \mathop{\sum }_{k=0}^{(p-1)/2}\frac{1}{2k-1}\binom{a}{k}\binom{-1-a}{k}\end{eqnarray}$$

$p^{2}$

We show that the transition matrix from the polytabloid basis to the web basis of the irreducible $\mathfrak{S}_{2n}$-representation of shape $(n,n)$ has nonnegative integer entries. This proves a conjecture of Russell and Tymoczko [Int. Math. Res. Not., 2019(5) (2019), 1479–1502].

We present an example of an isometric subspace of a metric space that has a greater metric dimension. We also show that the metric spaces of vector groups over the integers, defined by the generating set of unit vectors, cannot be resolved by a finite set. Bisectors in the spaces of vector groups, defined by the generating set consisting of unit vectors, are completely determined.

An elastic graph is a graph with an elasticity associated to each edge. It may be viewed as a network made out of ideal rubber bands. If the rubber bands are stretched on a target space there is an elastic energy. We characterize when a homotopy class of maps from one elastic graph to another is loosening, that is, decreases this elastic energy for all possible targets. This fits into a more general framework of energies for maps between graphs.

Quasi-Sturmian words, which are infinite words with factor complexity eventually $n+c$ share many properties with Sturmian words. In this article, we study the quasi-Sturmian colorings on regular trees. There are two different types, bounded and unbounded, of quasi-Sturmian colorings. We obtain an induction algorithm similar to Sturmian colorings. We distinguish them by the recurrence function.