We compute the large N limit of the partition function of the Euclidean Yang–Mills measure on orientable compact surfaces with genus $g\geqslant 1$ and non-orientable compact surfaces with genus $g\geqslant 2$, with structure group the unitary group ${\mathrm U}(N)$ or special unitary group ${\mathrm{SU}}(N)$. Our proofs are based on asymptotic representation theory: more specifically, we control the dimension and Casimir number of irreducible representations of ${\mathrm U}(N)$ and ${\mathrm{SU}}(N)$ when N tends to infinity. Our main technical tool, involving ‘almost flat’ Young diagram, makes rigorous the arguments used by Gross and Taylor (1993, Nuclear Phys. B 400(1–3) 181–208) in the setting of QCD, and in some cases, we recover formulae given by Douglas (1995, Quantum Field Theory and String Theory (Cargèse, 1993), Vol. 328 of NATO Advanced Science Institutes Series B: Physics, Plenum, New York, pp. 119–135) and Rusakov (1993, Phys. Lett. B 303(1) 95–98).

It is well known that for any integers k and g, there is a graph with chromatic number at least k and girth at least g. In 1960s, Erdös and Hajnal conjectured that for any k and g, there exists a number h(k,g), such that every graph with chromatic number at least h(k,g) contains a subgraph with chromatic number at least k and girth at least g. In 1977, Rödl proved the case when $g=4$, for arbitrary k. We prove the fractional chromatic number version of Rödl’s result.

Given a finite set $A \subseteq \mathbb{R}^d$, points $a_1,a_2,\dotsc,a_{\ell} \in A$ form an $\ell$-hole in A if they are the vertices of a convex polytope, which contains no points of A in its interior. We construct arbitrarily large point sets in general position in $\mathbb{R}^d$ having no holes of size $O(4^dd\log d)$ or more. This improves the previously known upper bound of order $d^{d+o(d)}$ due to Valtr. The basic version of our construction uses a certain type of equidistributed point sets, originating from numerical analysis, known as (t,m,s)-nets or (t,s)-sequences, yielding a bound of $2^{7d}$. The better bound is obtained using a variant of (t,m,s)-nets, obeying a relaxed equidistribution condition.

A (not necessarily proper) vertex colouring of a graph has clustering c if every monochromatic component has at most c vertices. We prove that planar graphs with maximum degree $\Delta$ are 3-colourable with clustering $O(\Delta^2)$. The previous best bound was $O(\Delta^{37})$. This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree $\Delta$ that exclude a fixed minor are 3-colourable with clustering $O(\Delta^5)$. The best previous bound for this result was exponential in $\Delta$.

Consider the following experiment: a deck with m copies of n different card types is randomly shuffled, and a guesser attempts to guess the cards sequentially as they are drawn. Each time a guess is made, some amount of ‘feedback’ is given. For example, one could tell the guesser the true identity of the card they just guessed (the complete feedback model) or they could be told nothing at all (the no feedback model). In this paper we explore a partial feedback model, where upon guessing a card, the guesser is only told whether or not their guess was correct. We show in this setting that, uniformly in n, at most $m+O(m^{3/4}\log m)$ cards can be guessed correctly in expectation. This resolves a question of Diaconis and Graham from 1981, where even the $m=2$ case was open.

Consider a random $n\times n$ zero-one matrix with ‘sparsity’ p, sampled according to one of the following two models: either every entry is independently taken to be one with probability p (the ‘Bernoulli’ model) or each row is independently uniformly sampled from the set of all length-n zero-one vectors with exactly pn ones (the ‘combinatorial’ model). We give simple proofs of the (essentially best-possible) fact that in both models, if $\min(p,1-p)\geq (1+\varepsilon)\log n/n$ for any constant $\varepsilon>0$, then our random matrix is nonsingular with probability $1-o(1)$. In the Bernoulli model, this fact was already well known, but in the combinatorial model this resolves a conjecture of Aigner-Horev and Person.

A graph G arrows a graph H if in every 2-edge-colouring of G there exists a monochromatic copy of H. Schelp had the idea that if the complete graph $K_n$ arrows a small graph H, then every ‘dense’ subgraph of $K_n$ also arrows H, and he outlined some problems in this direction. Our main result is in this spirit. We prove that for every sufficiently large n, if $n = 3t+r$ where $r \in \{0,1,2\}$ and G is an n-vertex graph with $\delta(G) \ge (3n-1)/4$, then for every 2-edge-colouring of G, either there are cycles of every length $\{3, 4, 5, \dots, 2t+r\}$ of the same colour, or there are cycles of every even length $\{4, 6, 8, \dots, 2t+2\}$ of the samecolour.

Our result is tight in the sense that no longer cycles (of length $>2t+r$) can be guaranteed and the minimum degree condition cannot be reduced. It also implies the conjecture of Schelp that for every sufficiently large n, every $(3t-1)$-vertex graph G with minimum degree larger than $3|V(G)|/4$ arrows the path $P_{2n}$ with 2n vertices. Moreover, it implies for sufficiently large n the conjecture by Benevides, Łuczak, Scott, Skokan and White that for $n=3t+r$ where $r \in \{0,1,2\}$ and every n-vertex graph G with $\delta(G) \ge 3n/4$, in each 2-edge-colouring of G there exists a monochromatic cycle of length at least $2t+r$.