Sean Longbrake

activity

20212026

most citedTree-degenerate graphs and nested dependent random choice

2 citations · 8 across the 16 of their papers we have counts for

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17 papers · 1 filter

math.CO2026

A proof of the rational exponents conjecture

David Conlon, Oliver Janzer, Tao Jiang +5

The rational exponents conjecture of Erdős and Simonovits asserts that for every rational $r \in [1, 2]$ there exists a graph $H$ such that $ex (n, H) = Θ (n^{r})$ . We present a pr…

math.CO2026

Rational exponents near 3/2

Tao Jiang, Sean Longbrake, Liana Yepremyan

Given a graph $H$ , the extremal number $e x (n, H)$ is the maximum number of edges in an $n$ -vertex graph not containing $H$ as a subgraph. The well-known rational exponents conjectur…

math.CO2026

On the generalized Turán number of complete bipartite graphs

Oliver Janzer, Sean Longbrake, Liana Yepremyan

For graphs $F$ and $H$ , the generalized Turán number $ex (n, F, H)$ denotes the maximum number of copies of $F$ in an $H$ -free graph on $n$ vertices. We prove that if $s\in \…

math.CO2026

Induced rational exponents near two

Tao Jiang, Sean Longbrake

Given a bipartite graph $H$ and a natural number $s$ , let $ex^{*} (n, H, s)$ denote the maximum number of edges in an $n$ -vertex graph that contains neither $K_{s, s}$ nor an in…

math.CO2026

Random Turán Problems for Graphs with a Vertex Complete to One Part

Sean Longbrake, Sam Spiro

Given a graph $F$ , the random Turán problem asks to determine the maximum number of edges in an $F$ -free subgraph of $G_{n, p}$ . Prior to this work, the only bipartite graphs $F$ wi…

math.CO2026

On the number of families avoiding a subposet

Tao Jiang, Sean Longbrake, Liana Yepremyan

In this paper we show that for any poset $P$ that is not an antichain, the number of induced $P$ -free families in the Boolean lattice $2^{[n]}$ is at most $ 2^{O(\mathrm{La}^*(n,P)…

math.CO2025

Regularization and asymmetric extremal numbers of subdivisions

Tao Jiang, Sean Longbrake

Given a real $μ \geq 1$ , a graph $H$ is $μ$ -almost-regular if $Δ (H) \leq μ δ (H)$ . The celebrated regularization theorem of Erdős and Simonovits states that for every real $0<\varepsil…

math.CO2025

Supersaturation via edge-gluing

Zihao Jin, Sean Longbrake, Liana Yepremyan

In 1984, Erdős and Simonovits conjectured the following: given a bipartite graph $H$ , there exist constants $β, C > 0$ such that any graph $G$ on $n$ vertices and $pn^2\geq C \math…

math.CO2025

Hamilton cycles in regular graphs perturbed by a random 2-factor

Cicely, Henderson, Sean Longbrake +2

In this paper, we prove that for each $d \geq 2$ , the union of a $d$ -regular graph with a uniformly random $2$ -factor on the same vertex set is Hamiltonian with high probability. T…

math.CO2025

Fractional hypergraph coloring

Margarita Akhmejanova, Sean Longbrake

We investigate proper $(a : b)$ -fractional colorings of $n$ -uniform hypergraphs, which generalize traditional integer colorings of graphs. Each vertex is assigned $b$ distinct colors…

math.CO2024★ 1 cited

On the number of H-free hypergraphs

Tao Jiang, Sean Longbrake

Two central problems in extremal combinatorics are concerned with estimating the number $e x (n, H)$ , the size of the largest $H$ -free hypergraph on $n$ vertices, and the number $forb…

math.CO2024★ 1 cited

Longest cycles in vertex-transitive and highly connected graphs

Carla Groenland, Sean Longbrake, Raphael Steiner +2

We present progress on three old conjectures about longest paths and cycles in graphs. The first pair of conjectures, due to Lovász from 1969 and Thomassen from 1978, respectively,…

math.CO2024

Some Cases of the Erdős-Lovász Tihany Conjecture for Claw-free Graphs

Sean Longbrake, Juvaria Tariq

The Erdős-Lovász Tihany Conjecture states that any $G$ with chromatic number $χ (G) = s + t - 1 > ω (G)$ , with $s, t \geq 2$ can be split into two vertex-disjoint subgraphs of chromat…

math.CO2024★ 2 cited

Tree Posets: Supersaturation, Enumeration, and Randomness

Tao Jiang, Sean Longbrake, Sam Spiro +1

We develop a powerful tool for embedding any tree poset $P$ of height $k$ in the Boolean lattice which allows us to solve several open problems in the area. We show that: * If $H$ …

math.CO2022★ 2 cited

Balanced supersaturation and Turan numbers in random graphs

Tao Jiang, Sean Longbrake

In a ground-breaking paper solving a conjecture of Erdős on the number of $n$ -vertex graphs not containing a given even cycle, Morris and Saxton \cite{MS} made a broad conjecture o…

math.CO2022★ 2 cited

Tree-degenerate graphs and nested dependent random choice

Tao Jiang, Sean Longbrake

The celebrated dependent random choice lemma states that in a bipartite graph an average vertex (weighted by its degree) has the property that almost all small subsets $S$ in its n…

math.CO2021

Bipartite-ness under smooth conditions

Tao Jiang, Sean Longbrake, Jie Ma

Given a family $F$ of bipartite graphs, the {\it Zarankiewicz number} $z (m, n, F)$ is the maximum number of edges in an $m$ by $n$ bipartite graph $G$ that does n…