Short Proof Research Articles

Equivariant formality of the isotropy action on (ℤ2 ⊕ ℤ2)-symmetric spaces

Compact symmetric spaces are probably one of the most prominent class of formal spaces, i.e. of spaces where the rational homotopy type is a formal consequence of the rational cohomology algebra. As a generalization, it is even known that their isotropy action is equivariantly formal. In this paper, we show that [Formula: see text]-symmetric spaces are equivariantly formal and formal in the sense of Sullivan, in particular. Moreover, we give a short alternative proof of equivariant formality in the case of symmetric spaces with our new approach.

Perimeter inequality under circular and Steiner symmetrisation: Geometric characterisation of extremals

We study the perimeter inequality under circular symmetrisation, and we provide a full geometric characterisation of equality cases. A careful inspection of the proof shows that a similar characterisation holds true also for the perimeter inequality under Steiner symmetrisation. Our result is based on a new short proof of the perimeter inequality under symmetrisation.

A Short ODE Proof of the Fundamental Theorem of Algebra

A Short ODE Proof of the Fundamental Theorem of Algebra

A Short Proof of Allard’s and Brakke’s Regularity Theorems

Abstract We give new short proofs of Allard’s regularity theorem for varifolds with bounded first variation and Brakke’s regularity theorem for integral Brakke flows with bounded forcing. They are based on a decay of flatness, following from weighted versions of the respective monotonicity formulas, together with a characterization of non-homogeneous blow-ups using the viscosity approach introduced by Savin.

The saturation number of K3,3

A graph G is called F-saturated if G does not contain F as a subgraph (not necessarily induced) but the addition of any missing edge to G creates a copy of F. The saturation number of F, denoted by sat(n,F), is the minimum number of edges in an n-vertex F-saturated graph. Determining the saturation number of complete bipartite graphs is one of the most important problems in the study of saturation numbers. The value of sat(n,K2,2) was shown to be ⌊3n−52⌋ by Ollmann, and a shorter proof was later given by Tuza. For K2,3, there has been a series of study aiming to determine sat(n,K2,3) over the years. This was finally achieved by Chen who confirmed a conjecture of Bohman, Fonoberova, and Pikhurko that sat(n,K2,3)=2n−3 for all n≥5. Pikhurko and Schmitt conjectured that sat(n,K3,3)=(3+o(1))n. In this paper, for n≥9, we give an upper bound of 3n−9 for sat(n,K3,3), and prove that 3n−9 is also a lower bound when the minimum degree of a K3,3-saturated graph is 2 or 5, where it is trivial when the minimum degree is greater than 5.

Characterizing the TTC rule via pair-efficiency: A short proof

In the object reallocation problem, Ekici (2023) showed that Top Trading Cycles (TTC) is the unique rule that is strategyproof, individual-rational, and pair-efficient. We provide a short proof of this characterization result.

Short Proof of the Asymptotic Confirmation of the Faudree-Lehel Conjecture

Given a simple graph $G$, the irregularity strength of $G$, denoted $s(G)$, is the least positive integer $k$ such that there is a weight assignment on edges $f: E(G) \to \{1,2,\dots, k\}$ for which each vertex weight $f^V(v):= \sum_{u: \{u,v\}\in E(G)} f(\{u,v\})$ is unique amongst all $v\in V(G)$. In 1987, Faudree and Lehel conjectured that there is a constant $c$ such that $s(G) \leq n/d + c$ for all $d$-regular graphs $G$ on $n$ vertices with $d>1$, whereas it is trivial that $s(G) \geq n/d$. In this short note we prove that the Faudree-Lehel Conjecture holds when $d \geq n^{0.8+\epsilon}$ for any fixed $\epsilon >0$, with a small additive constant $c=28$ for $n$ large enough. Furthermore, we confirm the conjecture asymptotically by proving that for any fixed $\beta\in(0,1/4)$ there is a constant $C$ such that for all $d$-regular graphs $G$, $s(G) \leq \frac{n}{d}(1+\frac{C}{d^{\beta}})+28$, extending and improving a recent result of Przybyło that $s(G) \leq \frac{n}{d}(1+ \frac{1}{\ln^{\epsilon/19}n})$ whenever $d\in [\ln^{1+\epsilon} n, n/\ln^{\epsilon}n]$ and $n$ is large enough.

From Abel’s Binomial Theorem to Cayley’s Tree Formula

We derive Abel’s generalization of the binomial theorem and use it to present a short proof of Cayley’s theorem on the number of trees on n labeled vertices.

A short proof of an index theorem, II

We introduce a slight modification of the usual equivariant KK -theory. We use this to give a KK -theoretical proof of an equivariant index theorem for Dirac–Schrödinger operators on a non-compact manifold of nowhere positive curvature. We incidentally show that the boundary of Dirac is Dirac; generalizing earlier work of Baum and coworkers, and a result of Higson and Roe.

Solving problems on generalized convex graphs via mim-width

A bipartite graph G=(A,B,E) is H-convex for some family of graphs H if there exists a graph H∈H with V(H)=A such that the neighbours in A of each b∈B induce a connected subgraph of H. Many NP-complete problems are polynomial-time solvable for H-convex graphs when H is the set of paths. The underlying reason is that the class has bounded mim-width. We extend this result to families of H-convex graphs where H is the set of cycles, or H is the set of trees with bounded maximum degree and a bounded number of vertices of degree at least 3. As a consequence, we strengthen many known results via one general and short proof. We also show that the mim-width of H-convex graphs is unbounded if H is the set of trees with arbitrarily large maximum degree or an arbitrarily large number of vertices of degree at least 3.

A Book Proof of the Middle Levels Theorem

We give a short constructive proof for the existence of a Hamilton cycle in the subgraph of the (2n+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2n+1)$$\\end{document}-dimensional hypercube induced by all vertices with exactly n or n+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n+1$$\\end{document} many 1s.

Boundedness of Wolff-type potentials and applications to PDEs

We provide a short proof of a sharp rearrangement estimate for a generalized version of a potential of Wolff–Havin–Maz’ya type. As a consequence, we prove a reduction principle for that integral operators, that is, a characterization of those rearrangement invariant spaces between which the potentials are bounded via a one-dimensional inequality of Hardy-type.Since the special case of the mentioned potential is known to control precisely very weak solutions to a broad class of quasilinear elliptic PDEs of non-standard growth, we infer the local regularity properties of the solutions in rearrangement invariant spaces for prescribed classes of data.

Exotic C⁎-algebras of geometric groups

We consider a new class of potentially exotic group C*-algebras C⁎(PFp⁎(G)) for a locally compact group G, and its connection with the class of potentially exotic group C*-algebras CLp⁎(G) introduced by Brown and Guentner. Surprisingly, these two classes of C*-algebras are intimately related. By exploiting this connection, we show CLp⁎(G)=C⁎(PFp⁎(G)) for p∈(2,∞), and the C*-algebras CLp⁎(G) are pairwise distinct for p∈(2,∞) when G belongs to a large class of nonamenable groups possessing the Haagerup property and either the rapid decay property or Kunze-Stein phenomenon by characterizing the positive definite functions that extend to positive linear functionals of CLp⁎(G) and C⁎(PFp⁎(G)). This greatly generalizes earlier results of Okayasu (see [30]) and the second author (see [40]) on the pairwise distinctness of CLp⁎(G) for 2<p<∞ when G is either a noncommutative free group or the group SL(2,R), respectively.As a byproduct of our techniques, we present two applications to the theory of unitary representations of a locally compact group G. Firstly, we give a short proof of the well-known Cowling-Haagerup-Howe Theorem, which presents sufficient condition implying the weak containment of a cyclic unitary representation of G in the left regular representation of G (see [14]). Also we give a near solution to a 1978 conjecture of Cowling stated in [10]. This conjecture of Cowling states if G is a Kunze-Stein group and π is a unitary representation of G with cyclic vector ξ such that the mapG∋s↦〈π(s)ξ,ξ〉 belongs to Lp(G) for some 2<p<∞, then Aπ⊆Lp(G). We show Bπ⊆Lp+ϵ(G) for every ϵ>0 (recall Aπ⊆Bπ).

Vanishing first cohomology and strong 1-boundedness for von Neumann algebras

We obtain a new proof of Shlyakhtenko's result which states that if G is a sofic, finitely presented group with vanishing first \ell^2 -Betti number, then L(G) is strongly 1-bounded. Our proof of this result adapts and simplifies Jung's technical arguments which showed strong 1-boundedness under certain conditions on the Fuglede–Kadison determinant of the matrix capturing the relations. Our proof also features a key idea due to Jung which involves an iterative estimate for the covering numbers of microstate spaces. We also use the works of Shlyakhtenko and Shalom to give a short proof that the von Neumann algebras of sofic groups with Property (T) are strongly 1 bounded, which is a special case of another result by the authors.

Induced paths in graphs without anticomplete cycles

Let us say a graph is Os-free, where s≥1 is an integer, if there do not exist s cycles of the graph that are pairwise vertex-disjoint and have no edges joining them. The structure of such graphs, even when s=2, is not well understood. For instance, until now we did not know how to test whether a graph is O2-free in polynomial time; and there was an open conjecture, due to Ngoc Khang Le, that O2-free graphs have only a polynomial number of induced paths.In this paper we prove Le's conjecture; indeed, we will show that for all s≥1, there exists c>0 such that every Os-free graph G has at most |G|c induced paths, where |G| is the number of vertices. This provides a poly-time algorithm to test if a graph is Os-free, for all fixed s.The proof has three parts. First, there is a short and beautiful proof, due to Le, that reduces the question to proving the same thing for graphs with no cycles of length four. Second, there is a recent result of Bonamy, Bonnet, Déprés, Esperet, Geniet, Hilaire, Thomassé and Wesolek, that in every Os-free graph G with no cycle of length four, there is a set of vertices that intersects every cycle, with size logarithmic in |G|. And third, there is an argument that uses the result of Bonamy et al. to deduce the theorem. The last is the main content of this paper.

On Rödl's Theorem for Cographs

A theorem of Rödl states that for every fixed $F$ and $\varepsilon&gt;0$ there is $\delta=\delta_F(\varepsilon)$ so that every induced $F$-free graph contains a vertex set of size $\delta n$ whose edge density is either at most $\varepsilon$ or at least $1-\varepsilon$. Rödl's proof relied on the regularity lemma, hence it supplied only a tower-type bound for $\delta$. Fox and Sudakov conjectured that $\delta$ can be made polynomial in $\varepsilon$, and a recent result of Fox, Nguyen, Scott and Seymour shows that this conjecture holds when $F=P_4$. In fact, they show that the same conclusion holds even if $G$ contains few copies of $P_4$. In this note we give a short proof of a more general statement.

A short proof of a strong form of the three dimensional Gaussian product inequality

We prove a strong form of the Gaussian product conjecture in dimension three. Our purely analytical proof simplifies previously known proofs based on combinatorial methods or computer-assisted methods, and allows us to solve the case of any triple of even positive integers which remained open so far.

A Short Simple Proof of Closedness of Convex Cones and Farkas’ Lemma

Proving that a finitely generated convex cone is closed is often considered the most difficult part of geometric proofs of Farkas’ lemma. We provide a short simple proof of this fact and (for completeness) derive Farkas’ lemma from it using well-known arguments.

Monochromatic exponential triples: An ultrafilter proof

We present a short ultrafilter proof of the existence of monochromatic exponential triples { a , b , b a } \{a, b, b^a\} in any finite coloring of the natural numbers. We then generalize the construction and prove a new result on the existence of infinite monochromatic exponential patterns.

A short proof of the Baker–Pixley theorem for classes

An important theorem by Baker and Pixley states that if [Formula: see text] is a finite algebra with a [Formula: see text]-ary near-unanimity term and [Formula: see text] is any function, then f is a term-function of [Formula: see text] if f preserves all subuniverses of [Formula: see text]. This result was generalized recently for (classes of) not necessarily finite algebras using sheaf-theoretic tools. In this work, we give a short model-theoretic proof of this generalization. Also, we apply the theorem to obtain a characterization of dual discriminator varieties, and to give necessary and sufficient conditions for a variety with a near-unanimity term to be congruence permutable.