In 1973, Erdős conjectured the existence of high girth (n,3,2)-Steiner systems. Recently, Glock, Kühn, Lo, and Osthus and independently Bohman and Warnke proved the approximate version of Erdős' conjecture. Just this year, Kwan, Sah, Sawhney, and Simkin proved Erdős' conjecture. As for Steiner systems with more general parameters, Glock, Kühn, Lo, and Osthus conjectured the existence of high girth (n,q,r)-Steiner systems. We prove the approximate version of their conjecture. This result follows from our general main results which concern finding perfect or almost perfect matchings in a hypergraph G avoiding a given set of submatchings (which we view as a hypergraph H where V(H)=E(G)). Our first main result is a common generalization of the classical theorems of Pippenger (for finding an almost perfect matching) and Ajtai, Komlós, Pintz, Spencer, and Szemerédi (for finding an independent set in girth five hypergraphs). More generally, we prove this for coloring and even list coloring, and also generalize this further to when H is a hypergraph with small codegrees (for which high girth designs is a specific instance). Indeed, the coloring version of our result even yields an almost partition of K_n^r into approximate high girth (n,q,r)-Steiner systems. Our main results also imply the existence of a perfect matching in a bipartite hypergraph where the parts have slightly unbalanced degrees. This has a number of applications; for example, it proves the existence of Δ pairwise disjoint list colorings in the setting of Kahn's theorem; it also proves asymptotic versions of various rainbow matching results in the sparse setting (where the number of times a color appears could be much smaller than the number of colors) and even the existence of many pairwise disjoint rainbow matchings in such circumstances.

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, Kühn, Lo, and Osthus proved that if the fractional relaxation of Nash-Williams' Conjecture holds for minimum degree cn for some constant c≥ 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= 7+√(21)/14≈ 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 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 3n/4 but no fractional K_3-decomposition, then G is close (in edit distance) to the join of two n/4-regular graphs each on n/2 vertices. We use this to prove that if a triangle-divisible graph G on n vertices has minimum degree close to 3n/4 but no K_3-decomposition, then G is close (in edit distance) to the join of two n/4-regular graphs each on n/2 vertices. In Part III, our final main result is a proof of Nash-Williams' Conjecture in full.

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≥ 4 by positing that every K_q-divisible graph on n vertices (for n large enough) with minimum degree at least (1-1/q+1)n has a K_q-decomposition. We disprove this conjecture for all q≥ 4; namely, we show that for each q≥ 4, there exists c > 1 such that there exist infinitely many K_q-divisible graphs G with minimum degree at least (1-1/c·(q+1))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 ε > 0 and every large enough integer q, there exist infinitely many K_q-divisible graphs G with minimum degree at least (1-1/(1+√(2)/2-ε)· (q+1))v(G) with no (fractional) K_q-decomposition.

We prove that if p greater than or equals n Superscript minus left parenthesis q minus 6 right parenthesis divided by 2 p >= n - ( q - 6 ) / 2 $p \geq n<^>{-(q-6)/2}$ , then asymptotically almost surely the binomial random q-uniform hypergraph script upper G Superscript left parenthesis q right parenthesis Baseline left parenthesis n comma p right parenthesis G ( q ) ( n , p ) $\mathcal{G}<^>{(q)}(n,p)$ contains an (n, q, 2)-Steiner system, provided n satisfies the necessary divisibility conditions.

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.

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.

In 2014, Keevash famously proved the existence of -Steiner systems as part of settling the Existence Conjecture of Combinatorial Designs (dating from the mid-1800s). In 2020, Glock, K & uuml;hn, and Osthus conjectured a minimum degree generalization: specifically that minimum -degree at least suffices to guarantee that every sufficiently large -divisible -uniform hypergraph on vertices admits a -decomposition (where is a constant that is allowed to depend on but not on ). The best-known progress on this conjecture is from the second proof of the existence conjecture by Glock, K & uuml;hn, Lo, and Osthus in 2016 who showed that suffices. The fractional relaxation of the conjecture is crucial to improving the bound; for that, only the slightly better bound of was known due to Barber, K & uuml;hn, Lo, Montgomery, and Osthus from 2017. Our main result is to prove that suffices for the fractional relaxation. Combined with the work of Henderson and Postle from 2025, this also shows that such -divisible hypergraphs admit -decompositions.

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.

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.

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≡ 1,3 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, Erdős conjectured that for each integer g, there exists a Steiner triple system on n vertices with girth at least g (provided that n≡ 1,3 6 is large enough compared to the fixed g). In 2021, Glock, Kühn, and Osthus conjectured the common generalization of these two conjectures, dubbing it the “Erdős 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, Kühn, Lo, and Osthus on Nash-Williams' Conjecture and the resolution of Erdős' 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).

We prove that up to two exceptions, every connected subcubic triangle-free graph has fractional chromatic number at most 11/4. This is tight unless further exceptional graphs are excluded, and improves the known bound on the fractional chromatic number of subcubic triangle-free planar graphs.

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, Kühn, 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 δ(G) hereafter) at least (1-Θ_r(1/q^r-1)) · 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, Kühn, Lo, and Osthus wherein they showed that δ(G)≥(1-c/q^2r)· 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 δ(G)≥(1-c/q^r-1 + o(1))· n guarantees a K_q^r-fractional decomposition. We prove that for every integer r≥ 2, there exists a real c>0 such that if a K_q^r-divisible r-graph G satisfies δ(G)≥max{ δ_K_q^r^* + ε, 1 -c/qr-1}· n, then G admits a K_q^r-decomposition for all large enough n, where δ_K_q^r^* denotes the fractional K_q^r-decomposition threshold. Combined with the fractional result above, this proves that (1-c/q^r-1 + o(1))· 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 Turán theory to prove the existence of many embeddings of absorbers which may be of independent interest.

We study k-star decompositions, that is, partitions of the edge set into disjoint stars with k edges, in the uniformly random dregular graph model Gn,d. Using the small subgraph conditioning method, we prove an existence result for such decompositions for all d, k such that d/2 < k < d/2 + max{1, 16 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 Gn,d for all d < 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 & aacute;sz et al. (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 2k2 + k < d, and (ii) when k is odd and k < d/2. (c) 2025 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

The famous List Colouring Conjecture from the 1970s states that for every graph G the chromatic index of G is equal to its list chromatic index. In 1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds asymptotically. Our main result is a local generalization of Kahn's theorem. More precisely, we show that, for a graph G with sufficiently large maximum degree Δ and minimum degree δ≥ln25⁡Δ, the following holds: for every assignment L of lists of colours to the edges of G, such that |L(e)|≥(1+o(1))⋅max⁡{deg⁡(u),deg⁡(v)} for each edge e=uv, there is an L-edge-colouring of G. Furthermore, Kahn showed that the List Colouring Conjecture holds asymptotically for linear, k-uniform hypergraphs, and recently Molloy generalized Kahn's original result to correspondence colouring as well as its hypergraph generalization. We prove local versions of all of these generalizations by showing a weighted version that simultaneously implies all of our results.

We provide a new short self-contained proof of the existence of K_q^r-absorbers. Combining this with the work of the first and third authors yields a proof of the Existence Conjecture for Combinatorial Designs that is not reliant on the construction of K_q^r-absorbers by Glock, Kühn, Lo, and Osthus.

A (vertex) colouring of graph is 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—no bound for acyclic list colouring locally planar graphs for any fixed number of colours was previously known.

In 1943, Hadwiger conjectured that every graph with no K_t minor is (t-1)-colorable for every t≥ 1. In the 1980s, Kostochka and Thomason independently proved that every graph with no K_t minor has average degree O(t√(log t)) and hence is O(t√(log t))-colorable. Recently, Norin, Song and the second author showed that every graph with no K_t minor is O(t(log t)^β)-colorable for every β > 1/4, making the first improvement on the order of magnitude of the O(t√(log t)) bound. The first main result of this paper is that every graph with no K_t minor is O(tloglog t)-colorable. This is a corollary of our main technical result that the chromatic number of a K_t-minor-free graph is bounded by O(t(1+f(G,t))) where f(G,t) is the maximum of χ(H)/a over all a≥t/√(log t) and K_a-minor-free subgraphs H of G that are small (i.e. O(alog^4 a) vertices). This has a number of interesting corollaries. First as mentioned, using the current best-known bounds on coloring small K_t-minor-free graphs, we show that K_t-minor-free graphs are O(tloglog t)-colorable. Second, it shows that proving Linear Hadwiger's Conjecture (that K_t-minor-free graphs are O(t)-colorable) reduces to proving it for small graphs. Third, we prove that K_t-minor-free graphs with clique number at most √(log t)/ (loglog t)^2 are O(t)-colorable. This implies our final corollary that Linear Hadwiger's Conjecture holds for K_r-free graphs for every fixed r. One key to proving the main theorem is a new standalone result that every K_t-minor-free graph of average degree d=Ω(t) has a subgraph on O(t log^3 t) vertices with average degree Ω(d).

In a fractional coloring, vertices of a graph are assigned measurable subsets of the real line and adjacent vertices receive disjoint subsets; the fractional chromatic number of a graph is at most k if it has a fractional coloring in which each vertex receives a subset of [0,1] of measure at least 1/k. We introduce and develop the theory of “fractional colorings with local demands” wherein each vertex “demands” a certain amount of color that is determined by local parameters such as its degree or the clique number of its neighborhood. This framework provides the natural setting in which to generalize degree-sequence type bounds on the independence number. Indeed, by Linear Programming Duality, all of the problems we study have an equivalent formulation as a problem concerning weighted independence numbers, and they often imply new bounds on the independence number.Our results and conjectures are inspired by many of the most classical results and important open problems concerning the independence number and the chromatic number, often simultaneously. We conjecture a local strengthening of both Shearer's bound on the independence number of triangle-free graphs and the fractional relaxation of Molloy's recent bound on their chromatic number, as well as a longstanding problem of Ajtai et al. on the independence number of Kr-free graphs and the fractional relaxations of Reed's ω,Δ,χ Conjecture and the Total Coloring Conjecture. We prove an approximate version of the first two, and we prove “local demands” versions of Vizing's Theorem and of some χ-boundedness results.

We prove the High Girth Existence Conjecture - the common generalization of the Existence Conjecture for Combinatorial Designs originating from the 1800s and Erdős' Conjecture from 1973 on the Existence of High Girth Steiner Triple Systems.

We prove that if p≥ n^-1/3+β for some β > 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 ≥ n^2/3+β for some β > 0 and d is even, then asymptotically almost surely the random d-regular graph G_n,d has a triangle decomposition provided 3 | d · 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≥ n^-1/q+0.5+β and leave of (q-2)n+O(1) edges, for K_q-decompositions of G_n,d with (q-1) | d and d≥ n^1-1/q+0.5+β provided q| d· n, and for fractional K_q-decompositions.