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Luke Postle

Publications and source records attributed to Luke Postle.

At least 19 recordsLinked to original sources
arXiv2026

A Proof of Nash-Williams' Conjecture

A central open question in extremal design theory is Nash-Williams' Conjecture from 1970 that every triangle-divisible graph on $n$ vertices (for $n$ large enough) with minimum degree at least $0.75 n$ has a triangle decomposition. In this paper, we prove this conjecture in full. In 2016, Barber, K\"{u}hn, Lo, and Osthus proved that if the fractional relaxation of Nash-Williams' Conjecture holds for minimum degree $cn$ for some constant $c\ge 0.75$, then Nash-Williams' Conjecture holds for any constant $c' > c$. The previously best-known bound on the fractional relaxation was due to Delcourt and Postle from 2021 with $c= \frac{7+\sqrt{21}}{14} \approx 0.82733$. This bound on the fractional relaxation has grown in importance over the years as it has been directly tied to bounds for a number of other problems in extremal design theory. This paper consists of three parts. In Part I, our first main result is a proof of the Fractional Nash-Williams' Conjecture: if $G$ is a graph on $n$ vertices with minimum degree at least $\frac{3n}{4}$, then $G$ has a fractional triangle decomposition. In Part II, our second main result is a Fractional Stability Theorem for Nash-Williams' Conjecture: if a graph $G$ on $n$ vertices has minimum degree close to $\frac{3n}{4}$ but no fractional $K_3$-decomposition, then $G$ is close (in edit distance) to the join of two $\frac{n}{4}$-regular graphs each on $\frac{n}{2}$ vertices. We use this to prove that if a triangle-divisible graph $G$ on $n$ vertices has minimum degree close to $\frac{3n}{4}$ but no $K_3$-decomposition, then $G$ is close (in edit distance) to the join of two $\frac{n}{4}$-regular graphs each on $\frac{n}{2}$ vertices. In Part III, our final main result is a proof of Nash-Williams' Conjecture in full.

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arXiv2025

On the Hypergraph Nash-Williams' Conjecture

In 2014, Keevash proved the existence of $(n,q,r)$-Steiner systems (equivalently $K_q^r$-decompositions of $K_n^r$) for all large enough $n$ satisfying the necessary divisibility conditions. In 2021, Glock, K\"uhn, and Osthus proposed a generalization of this result. Namely they conjectured a hypergraph version of Nash-Williams' Conjecture positing that if a $K_q^r$-divisible $r$-graph $G$ on $n$ vertices has minimum $(r-1)$-degree (denoted $\delta(G)$ hereafter) at least $\left(1-\Theta_r\left(\frac{1}{q^{r-1}}\right)\right) \cdot n$, then $G$ admits a $K_q^r$-decomposition. The best known progress on this conjecture dates to the second proof of the Existence Conjecture by Glock, K\"uhn, Lo, and Osthus wherein they showed that $\delta(G)\ge \left(1-\frac{c}{q^{2r}}\right)\cdot n$ suffices for large enough $n$, where $c$ is a constant depending on $r$ but not $q$. As for the fractional relaxation, the best known bound is due to Delcourt, Lesgourgues, and the second author, who proved that $\delta(G)\ge \left(1-\frac{c}{q^{r-1 + o(1)}}\right)\cdot n$ guarantees a $K_q^r$-fractional decomposition. We prove that for every integer $r\ge 2$, there exists a real $c>0$ such that if a $K_q^r$-divisible $r$-graph $G$ satisfies $\delta(G)\ge \max\left\{ \delta_{K_q^r}^* + \varepsilon,~~1 -\frac{c}{\binom{q}{r-1}} \right\} \cdot n$, then $G$ admits a $K_q^r$-decomposition for all large enough $n$, where $\delta_{K_q^r}^*$ denotes the fractional $K_q^r$-decomposition threshold. Combined with the fractional result above, this proves that $\left(1-\frac{c}{q^{r-1 + o(1)}}\right)\cdot n$ suffices for the Hypergraph Nash-Williams' Conjecture, approximately confirming the correct order of $q$. Our proof uses the newly developed method of refined absorption; we also develop a non-uniform Tur\'an theory to prove the existence of many embeddings of absorbers which may be of independent interest.

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arXiv2025

Refined Absorption: A New Proof of the Existence Conjecture and its Applications to Extremal and Probabilistic Design Theory

We discuss the recently developed method of refined absorption and how it is used to provide a new proof of the Existence Conjecture for combinatorial designs. This method can also be applied to resolve open problems in extremal and probabilistic design theory while providing a unified framework for these problems. Crucially, the main absorption theorem can be used as a "black-box" in these applications obviating the need to reprove the absorption step for each different setup.

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arXiv2025

Fractional Clique Decompositions of Dense Hypergraphs

In 2014, Keevash famously proved the existence of $(n,q,r)$-Steiner systems as part of settling the Existence Conjecture of Combinatorial Designs (dating from the mid-1800s). In 2020, Glock, K\"uhn, and Osthus conjectured a minimum degree generalization: specifically that minimum $(r-1)$-degree at least $(1-\frac{C}{q^{r-1}})n$ suffices to guarantee that every sufficiently large $K_q^r$-divisible $r$-uniform hypergraph on $n$ vertices admits a $K_q^r$-decomposition (where $C$ is a constant that is allowed to depend on $r$ but not $q$). The best-known progress on this conjecture is from the second proof of the Existence Conjecture by Glock, K\"uhn, Lo, and Osthus in 2016 who showed that $(1-\frac{C}{q^{2r}})n$ suffices. The fractional relaxation of the conjecture is crucial to improving the bound; for that, only the slightly better bound of $(1-\frac{C}{q^{2r-1}})n$ was known due to Barber, K\"uhn, Lo, Montgomery, and Osthus from 2017. Our main result is to prove that $(1-\frac{C}{q^{r-1+o(1)}})n$ suffices for the fractional relaxation. Combined with the work of R{\"o}dl, Schacht, Siggers, and Tokushige from 2007, this also shows that such hypergraphs admit approximate $K_q^r$-decompositions.

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arXiv2025

Beyond Nash-Williams: Counterexamples to Clique Decomposition Thresholds for All Cliques Larger than Triangles

A central open question in extremal design theory is Nash-Williams' Conjecture from 1970 that every $K_3$-divisible graph on $n$ vertices (for $n$ large enough) with minimum degree at least $3n/4$ has a $K_3$-decomposition. A folklore generalization of Nash-Williams' Conjecture extends this to all $q\ge 4$ by positing that every $K_q$-divisible graph on $n$ vertices (for $n$ large enough) with minimum degree at least $\left(1-\frac{1}{q+1}\right)n$ has a $K_q$-decomposition. We disprove this conjecture for all $q\ge 4$; namely, we show that for each $q\ge 4$, there exists $c > 1$ such that there exist infinitely many $K_q$-divisible graphs $G$ with minimum degree at least $\left(1-\frac{1}{c\cdot(q+1)}\right)v(G)$ and no $K_q$-decomposition; indeed we construct them admitting no fractional $K_q$-decomposition thus disproving the fractional relaxation of this conjecture. Our result also disproves the more general partite version. Indeed, we even show the folklore conjecture is off by a multiplicative factor by showing that for every $\varepsilon > 0$ and every large enough integer $q$, there exist infinitely many $K_q$-divisible graphs $G$ with minimum degree at least $\bigg(1-\frac{1}{\left(\frac{1+\sqrt{2}}{2}-\varepsilon\right)\cdot (q+1)}\bigg)v(G)$ with no (fractional) $K_q$-decomposition.

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arXiv2025

Erd\H{o}s meets Nash-Williams

In 1847, Kirkman proved that there exists a Steiner triple system on $n$ vertices (equivalently a triangle decomposition of the edges of $K_n$) whenever $n$ satisfies the necessary divisibility conditions (namely $n\equiv 1,3 \mod 6$). In 1970, Nash-Williams conjectured that every graph $G$ on $n$ vertices with minimum degree at least $3n/4$ (for $n$ large enough and satisfying the necessary divisibility conditions) has a triangle decomposition. In 1973, Erd\H{o}s conjectured that for each integer $g$, there exists a Steiner triple system on $n$ vertices with girth at least $g$ (provided that $n\equiv 1,3 \mod 6$ is large enough compared to the fixed $g$). In 2021, Glock, K\"uhn, and Osthus conjectured the common generalization of these two conjectures, dubbing it the ``Erd\H{o}s meets Nash-Williams' Conjecture''. In this paper, we reduce the combined conjecture to the fractional relaxation of the Nash-Williams' Conjecture. Combined with the best known fractional bound of Delcourt and Postle, this proves the combined conjecture above when $G$ has minimum degree at least $0.82733n$. We note that our result generalizes the seminal work of Barber, K\"uhn, Lo, and Osthus on Nash-Williams' Conjecture and the resolution of Erd\H{o}s' Conjecture by Kwan, Sah, Sawhney, and Simkin. Both previous proofs of those results used the method of iterative absorption. Our proof instead proceeds via the newly developed method of refined absorption (and hence provides new independent proofs of both results).

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arXiv2024

Acyclic List Colouring Locally Planar Graphs

A (vertex) colouring of graph is \emph{acyclic} if it contains no bicoloured cycle. In 1979, Borodin proved that planar graphs are acyclically 5-colourable. In 2010, Kawarabayashi and Mohar proved that locally planar graphs are acyclically 7-colourable. In 2002, Borodin, Fon-Der-Flaass, Kostochka, Raspaud, and Sopena proved that planar graphs are acyclically 7-list-colourable. We prove that locally planar graphs are acyclically 9-list-colourable\textemdash no bound for acyclic list colouring locally planar graphs for any fixed number of colours was previously known.

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arXiv2024

A Short Proof of the Existence of $K_q^r$-absorbers

We codify a short self-contained proof of the existence of $K_q^r$-absorbers implicit in Keevash's original proof of the Existence Conjecture. Combining this with the work of the first and third authors in yields a proof of the Existence Conjecture for Combinatorial Designs that is not reliant on the construction of $K_q^r$-absorbers by Glock, K\"uhn, Lo, and Osthus.

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arXiv2024

Improving the Caro-Wei bound and applications to Tur\'{a}n stability

We prove that if $G$ is a graph and $f(v) \leq 1/(d(v) + 1/2)$ for each $v\in V(G)$, then either $G$ has an independent set of size at least $\sum_{v\in V(G)}f(v)$ or $G$ contains a clique $K$ such that $\sum_{v\in K}f(v) > 1$. This result implies that for any $\sigma \leq 1/2$, if $G$ is a graph and every clique $K\subseteq V(G)$ has at most $(1 - \sigma)(|K| - \sigma)$ simplicial vertices, then $\alpha(G) \geq \sum_{v\in V(G)} 1 / (d(v) + 1 - \sigma)$. Letting $\sigma = 0$ implies the famous Caro-Wei Theorem, and letting $\sigma = 1/2$ implies that if fewer than half of the vertices in each clique of $G$ are simplicial, then $\alpha(G) \geq \sum_{v\in V(G)}1/(d(v) + 1/2)$, which is tight for the 5-cycle. When applied to the complement of a graph, this result implies the following new Tur\' an stability result. If $G$ is a $K_{r + 1}$-free graph with more than $(1 - 1/r)n^2/2 - n/4$ edges, then $G$ contains an independent set $I$ such that at least half of the vertices in $I$ are complete to $G - I$. Applying this stability result iteratively provides a new proof of the stability version of Tur\' an's Theorem in which $K_{r + 1}$-free graphs with close to the extremal number of edges are $r$-partite.

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arXiv2024

Refined Absorption: A New Proof of the Existence Conjecture

The study of combinatorial designs has a rich history spanning nearly two centuries. In a recent breakthrough, the notorious Existence Conjecture for Combinatorial Designs dating back to the 1800s was proved in full by Keevash via the method of randomized algebraic constructions. Subsequently Glock, K\"{u}hn, Lo, and Osthus provided an alternate purely combinatorial proof of the Existence Conjecture via the method of iterative absorption. We introduce a novel method of refined absorption for designs; here as our first application of the method we provide a new alternate proof of the Existence Conjecture (assuming the existence of $K_q^r$-absorbers by Glock, K\"{u}hn, Lo, and Osthus).

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arXiv2024

Clique Decompositions in Random Graphs via Refined Absorption

We prove that if $p\ge n^{-\frac{1}{3}+\beta}$ for some $\beta > 0$, then asymptotically almost surely the binomial random graph $G(n,p)$ has a $K_3$-packing containing all but at most $n + O(1)$ edges. Similarly, we prove that if $d \ge n^{\frac{2}{3}+\beta}$ for some $\beta > 0$ and $d$ is even, then asymptotically almost surely the random $d$-regular graph $G_{n,d}$ has a triangle decomposition provided $3 \mid d \cdot n$. We also show that $G(n,p)$ admits a fractional $K_3$-decomposition for such a value of $p$. We prove analogous versions for a $K_q$-packing of $G(n,p)$ with $p\ge n^{-\frac{1}{q+0.5}+\beta}$ and leave of $(q-2)n+O(1)$ edges, for $K_q$-decompositions of $G_{n,d}$ with $(q-1)~|~d$ and $d\ge n^{1-\frac{1}{q+0.5}+\beta}$ provided $q\mid d\cdot n$, and for fractional $K_q$-decompositions.

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arXiv2023

Exponentially Many Correspondence Colourings of Planar and Locally Planar Graphs

We show that there exists a constant $c > 0$ such that if $G$ is a planar graph with 5-correspondence assignment $(L,M)$, then $G$ has at least $2^{c\cdot v(G)}$ distinct $(L,M)$-colourings. This confirms a conjecture of Langhede and Thomassen. More broadly, we introduce a general method showing how hyperbolicity theorems for certain families of critical graphs can be used to derive lower bounds on the number of colourings of the associated class of planar graphs. Hence our main result follows from this method plus a technical theorem (that we proved in a previous paper) involving the hyperbolicity of graphs critical for $5$-correspondence colouring. We further demonstrate our method in the case of counting 3-correspondence colourings of planar graphs of girth at least five. Finally, we use these theorems to show analogous results hold in the case of counting 5-correspondence colourings of locally planar graphs, and counting 3-correspondence colourings of locally planar graphs of girth at least five.

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arXiv2023

Decomposing random regular graphs into stars

We study $k$-star decompositions, that is, partitions of the edge set into disjoint stars with $k$ edges, in the uniformly random $d$-regular graph model $\mathcal{G}_{n,d}$. Using the small subgraph conditioning method, we prove an existence result for such decompositions for all $d,k$ such that $d/2 < k \leq d/2 + \max\{1,\frac{1}{6}\log d\}$. More generally, we give a sufficient existence condition that can be checked numerically for any given values of $d$ and $k$. Complementary negative results are obtained using the independence ratio of random regular graphs. Our results establish an existence threshold for $k$-star decompositions in $\mathcal{G}_{n,d}$ for all $d\leq 100$ and $k > d/2$. For smaller values of $k$, the connection between $k$-star decompositions and $\beta$-orientations allows us to apply results of Thomassen (2012) and Lov\'asz, Thomassen, Wu and Zhang (2013). We prove that random $d$-regular graphs satisfy their assumptions with high probability, thus establishing a.a.s. existence of $k$-star decompositions (i) when $2k^2+k\leq d$, and (ii) when $k$ is odd and $k < d/2$.

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arXiv2023

Hyperbolicity Theorems for Correspondence Colouring

We generalize a framework of list colouring results to correspondence colouring. Correspondence colouring is a generalization of list colouring wherein we localize the meaning of the colours available to each vertex. As pointed out by Dvo\v{r}\'ak and Postle, both of Thomassen's theorems on the 5-choosability of planar graphs and 3-choosability of planar graphs of girth at least five carry over to the correspondence colouring setting. In this paper, we show that the family of graphs that are critical for 5-correspondence colouring as well as the family of graphs of girth at least five that are critical for 3-correspondence colouring form hyperbolic families. Analogous results for list colouring were shown by Postle and Thomas and by Dvo\v{r}\'ak and Kawarabayashi, respectively. Using results on hyperbolic families due to Postle and Thomas, we show further that this implies that locally planar graphs are 5-correspondence colourable; and, using results of Dvo\v{r}\'ak and Kawarabayashi, that there exist linear-time algorithms for the decidability of 5-correspondence colouring for embedded graphs. We show analogous results for 3-correspondence colouring graphs of girth at least five.

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arXiv2022

On generalized Ramsey numbers in the non-integral regime

A $(p,q)$-coloring of a graph $G$ is an edge-coloring of $G$ such that every $p$-clique receives at least $q$ colors. In 1975, Erd\H{o}s and Shelah introduced the generalized Ramsey number $f(n,p,q)$ which is the minimum number of colors needed in a $(p,q)$-coloring of $K_n$. In 1997, Erd\H{o}s and Gy\'arf\'as showed that $f(n,p,q)$ is at most a constant times $n^{\frac{p-2}{\binom{p}{2} - q + 1}}$. Very recently the first author, Dudek, and English improved this bound by a factor of $\log n^{\frac{-1}{\binom{p}{2} - q + 1}} $ for all $q \le \frac{p^2 - 26p + 55}{4}$, and they ask if this improvement could hold for a wider range of $q$. We answer this in the affirmative for the entire non-integral regime, that is, for all integers $p, q$ with $p-2$ not divisible by $\binom{p}{2} - q + 1$. Furthermore, we provide a simultaneous three-way generalization as follows: where $p$-clique is replaced by any fixed graph $F$ (with $|V(F)|-2$ not divisible by $|E(F)| - q + 1$); to list coloring; and to $k$-uniform hypergraphs. Our results are a new application of the Forbidden Submatching Method of the second and fourth authors.

math.CO↗
arXiv2022

The limit in the $(k+2, k)$-Problem of Brown, Erd\H{o}s and S\'os exists for all $k\geq 2$

Let $f^{(r)}(n;s,k)$ be the maximum number of edges of an $r$-uniform hypergraph on~$n$ vertices not containing a subgraph with $k$~edges and at most $s$~vertices. In 1973, Brown, Erd\H{o}s and S\'os conjectured that the limit $$\lim_{n\to \infty} n^{-2} f^{(3)}(n;k+2,k)$$ exists for all positive integers $k\ge 2$. They proved this for $k=2$. In 2019, Glock proved this for $k=3$ and determined the limit. Quite recently, Glock, Joos, Kim, K\"{u}hn, Lichev and Pikhurko proved this for $k=4$ and determined the limit; we combine their work with a new reduction to fully resolve the conjecture by proving that indeed the limit exists for all positive integers $k\ge 2$.

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