Sean Longbrake

Ph.D. Student
Emory University - Department of Mathematics

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Abstract:Given a graph $H$, the extremal number $ex(n,H)$ is the maximum number of edges in an $n$-vertex graph not containing $H$ as a subgraph. The well-known rational exponents conjecture of Erdős and Simonovits states that for any rational $\gamma\in (1,2)$ there exists a single bipartite graph $H$ satisfying $ex(n,H)=\Theta(n^\gamma)$. Among other results, the conjecture has been verified for all $\gamma=1+a/b$, where $b>a^2$, by Jiang and Qiu and for all $\gamma=2-a/b$, where $b>\max\{a, (a-1)^2\}$, by Conlon and Janzer.
In this paper, we establish the rational exponents conjecture for many $\gamma$ near the center of the interval, namely, for all $\gamma=1+\frac{rt-1}{2rt+2r}$, where $r,t$ are natural numbers satisfying $t\geq 2$, $r\geq 2t+3$.

Abstract:For graphs $F$ and $H$, the generalized Turán number $\mathrm{ex}(n,F,H)$ denotes the maximum number of copies of $F$ in an $H$-free graph on $n$ vertices. We prove that if $s\in \{2,3\}$, $s< a\leq b$ and $t$ is sufficiently large, then $\mathrm{ex}(n,K_{a,b},K_{s,t})=\Theta(n^s)$. The $s=2$, $a=b=3$ case of this result answers a question of Spiro.
Proving another conjecture of Spiro, we show that for every graph $F$ with at least one edge, there exist infinitely many real numbers $r$ such that $\mathrm{ex}(n,F,H)=\Theta(n^r)$ holds for some graph $H$.

Abstract:Given a bipartite graph $H$ and a natural number $s$, let $\mathrm{ex}^*(n,H,s)$ denote the maximum number of edges in an $n$-vertex graph that contains neither $K_{s,s}$ nor an induced copy of $H$. Hunter, Milojević, Sudakov, and Tomon conjectured that $\mathrm{ex}^*(n,H,s)=O_{H,s}(\mathrm{ex}(n,H))$ whenever $H$ is connected. Motivated by this conjecture and the rational exponents conjecture, Dong, Gao, Li, and Liu conjectured that for every rational $r\in (1,2)$ there is a bipartite graph $H$ and an $s_0$ such that $\mathrm{ex}^*(n,H,s)=\Theta(n^r)$ for all $s\geq s_0$.
We prove that the latter conjecture holds for all rationals $r=2-a/b$, where $a,b\in\mathbb{N}$ satisfy $b\geq \max\{a,(a-1)^2\}$. Our result extends a well-known result of Conlon and Janzer to the induced setting and adds more evidence to support the former conjecture.

Abstract: 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 $F$ with known tight bounds included certain classes of complete bipartite graphs and theta graphs. We greatly expand upon these examples by proving tight bounds for a number of bipartite graphs which have a vertex complete to one part. We also prove new general upper bounds for this problem which in many cases do significantly better than the only previous known general upper bound due to Jiang and Longbrake. Our proofs utilize dependent random choice together with the recent technique of balanced vertex supersaturation in conjunction with hypergraph containers.

Abstract: 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))}$, where $\mathrm{La}^*(n,P)$ denotes the the largest size of an induced $P$-free subfamily of $2^{[n]}$. We also obtain related supersaturation results.

Abstract: In 1984, Erdős and Simonovits asked the following: given a bipartite graph $H$, do there exist constants $0 \leq α< 1$ and $β, C > 0$ such that any graph $G$ on $n$ vertices and $pn^2\geq C n^{1+ α}$ edges contains at least $βn^{\mathrm{v}(H)} p^{\mathrm{e}(H)}$ copies of $H$? We show that edge-gluing preserves the satisfiability of this conjecture under some mild symmetry conditions. Namely, if two graphs $H_1$ and $H_2$ satisfy this conjecture, and if furthermore, gluing them along a fixed edge produces a unique graph then the resulting graph satisfies the conjecture as well. We also show that if $H$ satisfies the conjecture then if we glue several copies of (labeled) $H$ along the same labeled copy of a subforest of $H$ then the resulting graph also satisfies the conjecture. We also show that Zarankiewicz numbers are additive in the order of magnitude under gluing edges. Indeed, for a (signed) bipartite graph $H$ with parts coloured $+$ and $-$, recall $z(m,n, H)$ is the maximum number of edges in a signed bipartite graph $G$ with $+$ side being of size $m$ and $-$ side being of size $n$ such that $G$ does not contain a copy of $H$ with $+$ side embedded in the $+$ side of $G$. We show that for any two (signed) bipartite graphs $H_1$ and $H_2$ if we glue them along an edge preserving the sign of the edge then the resulting graph $H$ satisfies $z(m,n, H) = Θ(z(m,n, H_1) + z(m,n, H_2))$.

Abstract: Given a real $μ\geq 1$, a graph $H$ is $μ$-almost-regular if $Δ(H)\leq μδ(H)$. The celebrated regularization theorem of Erdős and Simonovits states that for every real $0<\varepsilon<1$ there exists a real $μ=μ(\varepsilon)$ such that every $n$-vertex graph $G$ with $Ω(n^{1+\varepsilon})$ edges contains an $m$-vertex $μ$-almost-regular subgraph $H$ with $Ω(m^{1+\varepsilon})$ edges for some $n^{\varepsilon\frac{1-\varepsilon}{1+\varepsilon}}\leq m\leq n$. We develop an enhanced version of it in which the subgraph $H$ also has average degree at least $Ω(\frac{d(G)}{\log n})$, where $d(G)$ is the average degree of $G$. We then give a bipartite analogue of the enhanced regularization theorem. Using the bipartite regularization theorem, we establish upper bounds on the maximum number of edges in a bipartite graph with part sizes $m$ and $n$ that does not contain a $2k$-subdivision of $K_{s,t}$ or $2k$-multi-subdivisions of $K_p$, thus extending the corresponding work of Janzer to the bipartite setting for even subdivisions. We show these upper bounds are tight up to a constant factor for infinitely many pairs $(m,n)$. The problem for estimating the maximum number of edges in a bipartite graph with part sizes $m$ and $n$ that does not contain a $(2k+1)$-subdivision of $K_{s,t}$ remains open.

Abstract: 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. This resolves a conjecture by Draganić and Keevash for all values of $d$.

Abstract: 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 from a set of $a$ colors, and an edge is properly colored if no single color is shared by all vertices of the edge. A hypergraph is $(a:b)$-colorable if every edge is properly colored. We prove that for any $2\leq b\leq a-2\leq n/\ln n$, every $n$-uniform hypergraph $H$ with $ |E(H)| \leq (ab^3)^{-1/2}\left(\frac{n}{\log n}\right)^{1/2} \left(\frac{a}{b}\right)^{n-1} $ is proper $(a:b)$-colorable. We also address specific cases, including $(a:a-1)$-colorability.

Forum of Math, Sigma. Vol 14, (2026). arXiv

Abstract: Two central problems in extremal combinatorics are concerned with estimating the number ${\rm ex}(n,\mathcal{H})$, the size of the largest $\mathcal{H}$-free hypergraph on $n$ vertices, and the number ${\rm forb}(n,\mathcal{H})$ of $\mathcal{H}$-free hypergraph on $n$ vertices. While it is known that ${\rm forb}(n,\mathcal{H})=2^{(1+o(1)){\rm ex}(n,\mathcal{H})}$ for $k$-uniform hypergraphs that are not $k$-partite, estimates for hypergraphs that are $k$-partite (or degenerate) are not nearly as tight. In a recent breakthrough, Ferber, McKinley, and Samotij proved that for many degenerate hypergraphs $\mathcal{H}$, ${\rm forb}(n, \mathcal{H}) = 2^{O({\rm ex}(n, \mathcal{H}))}$. However, there are few known instances of degenerate hypergraphs $\mathcal{H}$ for which ${\rm forb}(n,\mathcal{H})=2^{(1+o(1)){\rm ex}(n,\mathcal{H})}$ holds. In this paper, we show that ${\rm forb}n,\mathcal{H})=2^{(1+o(1)){\rm ex}(n,\mathcal{H})}$ holds for a wide class of degenerate hypergraphs known as $2$-contractible hypertrees. This is the first known infinite family of degenerate hypergraphs $\mathcal{H}$ for which ${\rm forb}(n,\mathcal{H})=2^{(1+o(1)){\rm ex}(n,\mathcal{H})}$ holds. As a corollary of our main results, we obtain a surprisingly sharp estimate of ${\rm forb}(n,C^{(k)}_\ell)=2^{(\lfloor\frac{\ell-1}{2}\rfloor+o(1))\binom{n}{k-1}}$ for the $k$-uniform linear $\ell$-cycle, for all pairs $k\geq 5, \ell\geq 3$, thus settling a question of Balogh, Narayanan, and Skokan affirmatively for all $k\geq 5, \ell\geq 3$. Our methods also lead to some related sharp results on the corresponding random Turán problem.

Canadian Journal of Mathematics, (2025). arXiv

Abstract: 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 $\mathcal{F}$ is a family in $\mathcal{B}_n$ with $|\mathcal{F}|\ge (q-1+\varepsilon){n\choose \lfloor n/2\rfloor}$ for some $q\ge k$, then $\mathcal{F}$ contains on the order of as many induced copies of $P$ as is contained in the $q$ middle layers of the Boolean lattice. This generalizes results of Bukh and Boehnlein and Jiang which guaranteed a single such copy in non-induced and induced settings respectively.
• The number of induced $P$-free families of $\mathcal{B}_n$ is $2^{(k-1+o(1)){n\choose \lfloor n/2\rfloor}}$, strengthening recent independent work of Balogh, Garcia, Wigal who obtained the same bounds in the non-induced setting.
• The largest induced $P$-free subset of a $p$-random subset of $\mathcal{B}_n$ for $p\gg n^{-1}$ has size at most $(k-1+o(1))p{n\choose \lfloor n/2\rfloor}$, generalizing previous work of Balogh, Mycroft, and Treglown and of Collares and Morris for the case when $P$ is a chain.
All three results are asymptotically tight and give affirmative answers to general conjectures of Gerbner, Nagy, Patkós, and Vizer in the case of tree posets.

Bulletin of the London Mathematical Society, Vol 57, no 10. (2025).arXiv

Abstract: 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, states that all connected vertex-transitive graphs contain a Hamiltonian path, and that all sufficiently large such graphs even contain a Hamiltonian cycle. The third conjecture, due to Smith from 1984, states that for $r\ge 2$ in every $r$-connected graph any two longest cycles intersect in at least $r$ vertices. In this paper, we prove a new lemma about the intersection of longest cycles in a graph which can be used to improve the best known bounds towards all the aforementioned conjectures: First, we show that every connected vertex-transitive graph on $n\geq 3$ vertices contains a cycle (and hence path) of length at least $\Omega(n^{13/21})$, improving on $\Omega(n^{3/5})$ from De Vos 2023. Second, we show that in every $r$-connected graph with $r\geq 2$, any two longest cycles meet in at least $\Omega(r^{5/8})$ vertices, improving on $\Omega(r^{3/5})$ from Chen, Faudree and Gould 1998. Our proof combines combinatorial arguments, computer-search and linear programming.

Discrete Mathematics, Vol 349, Iss 3. (2026).arXiv

Abstract:The Erdős-Lovász Tihany Conjecture states that any $G$ with chromatic number $\chi(G) = s + t - 1 > \omega(G)$, with $s,t \geq 2$ can be split into two vertex-disjoint subgraphs of chromatic number $s, t$ respectively. We prove this conjecture for pairs $(s, t)$ if $t \leq s + 2$, whenever $G$ has a $K_s$, and for pairs $(s, t)$ if $t \leq 4 s - 3$, whenever $G$ contains a $K_s$ and is claw-free. We also prove the Erdős-Lovász Tihany Conjecture for the pair $(3, 10)$ for claw-free graphs.

Advances in Combinatorics, 2024:3 arXiv

Abstract: 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 on so-called balanced supersaturation property of a bipartite graph $H$. Ferber, McKinley, and Samotij \cite{FMS} established a weaker version of this conjecture and applied it to derive far-reaching results on the enumeration problem of $H$-free graphs.
In this paper, we show that Morris and Saxton's conjecture holds under a very mild assumption about $H$, which is widely believed to hold whenever $H$ contains a cycle. We then use our theorem to obtain enumeration results and general upper bounds on the Turán number of a bipartite $H$ in the random graph $G(n,p)$, the latter being first of its kind.

SIAM Journal of Discrete Mathematics. Vol 37. Iss. 3 (2023). arXiv

Abstract: 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 neighborhood has common neighborhood almost as large as in the random graph of the same edge-density. Two well-known applications of the lemma are as follows. The first is a theorem of Füredi and of Alon, Krivelevich, and Sudakov showing that the maximum number of edges in an $n$-vertex graph not containing a fixed bipartite graph with maximum degree at most $r$ on one side is $O(n^{2-1/r})$. This was recently extended by Grzesik, Janzer and Nagy to the family of so-called $(r,t)$-blowups of a tree. A second application is a theorem of Conlon, Fox, and Sudakov, confirming a special case of a conjecture of Erdős and Simonovits and of Sidorenko, showing that if $H$ is a bipartite graph that contains a vertex complete to the other part and $G$ is a graph then the probability that the uniform random mapping from $V(H)$ to $V(G)$ is a homomorphismis at least $\left[\frac{2|E(G)|}{|V(G)|^2}\right]^{|E(H)|}$.
In this note, we introduce a nested variant of the dependent random choice lemma, which might be of independent interest. We then apply it to obtain a common extension of the theorem of Conlon, Fox, and Sudakov and the theorem of Grzesik, Janzer, and Nagy, regarding Turán and Sidorenko properties of so-called tree-degenerate graphs.

Combinatorics, Probability, and Computing. 31 (2022), 333-344. arXiv

Abstract: Given a family $\mathcal{F}$ of bipartite graphs, the {\it Zarankiewicz number} $z(m,n,\mathcal{F})$ is the maximum number of edges in an $m$ by $n$ bipartite graph $G$ that does not contain any member of $\mathcal{F}$ as a subgraph (such $G$ is called {\it $\mathcal{F}$-free}). For $1\leq \beta<\alpha<2$, a family $\mathcal{F}$ of bipartite graphs is $(\alpha,\beta)$-{\it smooth} if for some $\rho>0$ and every $m\leq n$, $z(m,n,\mathcal{F})=\rho m n^{\alpha-1}+O(n^\beta)$. Motivated by their work on a conjecture of Erdős and Simonovits on compactness and a classic result of Andrásfai, Erdős and Sós, Allen, Keevash, Sudakov and Verstraëte proved that for any $(\alpha,\beta)$-smooth family $\mathcal{F}$, there exists $k_0$ such that for all odd $k\geq k_0$ and sufficiently large $n$, any $n$-vertex $\mathcal{F}\cup\{C_k\}$-free graph with minimum degree at least $\rho(\frac{2n}{5}+o(n))^{\alpha-1}$ is bipartite.
In this paper, we strengthen their result by showing that for every real $\delta>0$, there exists $k_0$ such that for all odd $k\geq k_0$ and sufficiently large $n$, any $n$-vertex $\mathcal{F}\cup\{C_k\}$-free graph with minimum degree at least $\delta n^{\alpha-1}$ is bipartite. Furthermore, our result holds under a more relaxed notion of smoothness, which include the families $\mathcal{F}$ consisting of the single graph $K_{s,t}$ when $t\gg s$. We also prove an analogous result for $C_{2\ell}$-free graphs for every $\ell\geq 2$, which complements a result of Keevash, Sudakov and Verstraëte.