A central theme in extremal graph theory is the supersaturation problem, which investigates the minimum number of copies of a target subgraph forced by prescribed edge conditions. This line of research goes back to Rademacher and Erd\H{o}s for triangles, and was later extended to cliques by Lov\'asz and Simonovits in the regime above the Tur\'an threshold. Mubayi further extended this theory to color-critical graphs. Below the Tur\'an threshold, a closely related existence-threshold phenomenon arises in the non-$p$-partite setting: a classical result of Brouwer shows that, for $n\ge 2p+1$, every $n$-vertex non-$p$-partite $K_{p+1}$-free graph has at most $e(T_{n,p})-\lfloor n/p\rfloor+1$ edges. Motivated by this threshold, we investigate a sharp clique-counting problem below the Tur\'an threshold under the non-$p$-partite assumption. Let $p\ge 2$ and $s\ge 1$ be fixed integers. Let $Y_{n,p,s}$ be the graph obtained from $T_{n,p}$ by adding an edge inside a largest part and deleting all but $s$ of the edges from one endpoint of this new edge to a smallest part. Then $e(Y_{n,p,s})=e(T_{n,p})-\lfloor n/p\rfloor+s+1$. We prove that, for all sufficiently large $n$, every $n$-vertex non-$p$-partite graph $G$ with $e(G)\ge e(Y_{n,p,s})$ contains at least as many copies of $K_{p+1}$ as $Y_{n,p,s}$ does. The bound is sharp, as it is attained by the construction $Y_{n,p,s}$. Thus our result provides the exact clique-counting analogue of Brouwer's threshold for non-$p$-partite $K_{p+1}$-free graphs.
A {\it threshold graph} is a graph that can be constructed from the one-vertex graph by repeatedly adding either a dominating vertex or an isolated vertex. Motivated by an induced Ramsey-type problem for this class, we define $r'_2(s)$ to be the minimum integer $n$ such that every $n$-vertex graph contains an induced threshold graph on $s$ vertices. We establish exponential upper and lower bounds for $r'_2(s)$ and determine its exact values for $s\in\{3,4,5,6\}$. To study this problem from an edge-coloring perspective, we use the notion of an orderable coloring, introduced by Richer [{\it J. Combin. Theory Ser. B}, 80(1) (2000), 172--177]. An edge-colored graph is {\it orderable} if its vertices can be ordered so that, for each vertex, all edges from it to later vertices have the same color. Equivalently, $r'_2(s)$ is the minimum $n$ such that every $2$-edge-coloring of $K_n$ contains an orderable $K_s$. We also determine the exact value of the unordered canonical Ramsey number $CR(s, 3)$ for all $s \ge 3$, where $CR(s,3)$ denotes the minimum integer $n$ such that every edge-coloring of $K_n$ contains either an orderable $K_s$ or a rainbow $K_3$. More generally, for graphs $G$ and $H$, we study $r'_2(G)$, the corresponding $2$-color Ramsey number for an orderable $G$, and $CR(G,H)$, where the alternative is a rainbow $H$. For complete bipartite graphs, we prove that for every fixed $s$, $r'_2(K_{s,t}) = CR(K_{s,t}, K_3)= \left(\frac{2^s}{s+1}+o(1)\right)t$ as $t\to\infty$. For $s\in \{2,3\}$, we further determine the exact values of these parameters for infinitely many $t$, using constructions arising from strongly regular graphs, Hadamard matrices and conference matrices.
For an $F$-free graph $G$, a non-edge is $F$-saturating if adding it to $G$ creates a copy of $F$. We denote by $f_{p+1}(n,m)$ the minimum number of $K_{p+1}$-saturating non-edges in a $K_{p+1}$-free $n$-vertex graph with $m$ edges. Erd\H{o}s and Tuza conjectured that $f_4\left(n,\mathrm{ex}(n,K_3)+ 1\right)= (1 + o(1)) \frac{n^2}{16}$. Balogh and Liu (JCTB, 2014) disproved this conjecture and determined the asymptotic value of $f_4(n,\mathrm{ex}(n,K_3)+1)$. He, Ma, Ma and Ye (JCTB, 2023) later determined $f_{p+1}(n,\mathrm{ex}(n,K_p)+1)$ asymptotically for every $p\ge 3$, and asked for the value of $f_{p+1}(n,m)$ for all $\mathrm{ex}(n,K_p)+1\le m\le \mathrm{ex}(n,K_{p+1})$ and every $p\ge 3$. In this paper, we answer their question asymptotically for all $\mathrm{ex}(n,K_p)+1\le m\le \mathrm{ex}(n,K_{p+1})$ and every $p\ge 3$. We also determine the exact value of $f_3(n,m)$ for all $0\le m\le \mathrm{ex}(n,K_3)$ by a different method.
The P\'{o}sa-Seymour conjecture establishes the minimum degree threshold required to guarantee the presence of the $k$th power of a Hamilton cycle in a graph. Following numerous partial results, Koml\'{o}s, S\'{a}rk\"{o}zy, and Szemer\'{e}di confirmed the conjecture holds for all sufficiently large graphs. Treglown later conjectured the analogous minimum semi-degree threshold for forcing the $k$th power of a Hamilton cycle in a digraph. Subsequently, DeBiasio et al. proposed a conjecture on the minimum total degree threshold for the same problem. In this paper we settle the conjecture of DeBiasio et al. for $k=2$. Specifically, we prove that every sufficiently large $n$-vertex digraph with minimum total degree at least $8n/5-c$ contains the square of a Hamilton cycle, where $c=2$ if $n\equiv2,4\pmod 5$, and $c=1$ otherwise.
For a graph $G$, let $h(G)$ be the minimum cardinality of a vertex set meeting every maximum independent set of $G$. We establish two complementary reduction principles for the Bollob\'as--Erd\H{o}s--Tuza conjecture: the conjecture for arbitrary graphs is equivalent to its restriction to regular graphs of any fixed positive linear degree, and, within every hereditary graph class, a uniform sublinear bound is equivalent to a sublinear bound on graphs of every fixed positive linear vertex connectivity. We prove the sharp general estimate \[ h(G)\le \left\lfloor\frac{|V(G)|}{2\alpha(G)+\delta(G)-|V(G)|}\right\rfloor \] whenever the denominator is positive, with equality for balanced complete multipartite graphs. Consequently, every $3$-colorable graph of order $n$ with $\kappa(G)\ge\rho n$ and $\rho>1/3$ has a hitting set of size at most $\lfloor(\rho-1/3)^{-1}\rfloor$; direct use of a $3$-coloring improves this to $6$ when $\kappa(G)>4n/9$ and to the sharp bound $3$ when $\kappa(G)>n/2$. For dense regular graphs with independence ratio greater than $1/4$, we obtain a logarithmic bound, while constructions with linear degree and linear independence number show that $h(G)=\Omega(\sqrt n)$ can still occur. We also prove a logarithmic bound for near-regular $3$-colorable graphs and exhibit a critical family at connectivity $n/3$ that explains the limitations of the degree-surplus and degree-ratio methods.
The $\text{bk}(G)$ of a graph $G$ is the maximum number of triangles sharing a common edge. Motivated by a classical conjecture of Erd\H{o}s, spectral lower bounds for the booksize have received considerable attention. For a positive divisor $s$ of $m-1$ with $\frac{m-1}{s}\ge2$, let $S_{m,s}^{+}$ be obtained from $K_{s,\frac{m-1}{s}}$ by adding one edge inside the part of order $\frac{m-1}{s}$. Zhai et al. proved that, apart from this explicit family, every $m$-edge non-bipartite graph satisfying $\rho(G)^2\ge m-1+\frac{2}{\rho(G)-1}$ has booksize greater than $\frac{1}{240}\sqrt{m}$, and they asked for the best possible constant. We answer this question asymptotically. For every $0<\varepsilon<\frac{1}{4}$ and all sufficiently large $m$, every $m$-edge non-bipartite graph $G$ without isolated vertices satisfying the same spectral condition either is isomorphic to $S_{m,s}^{+}$ for some such integer $s$, or satisfies $\text{bk}(G)>\left(\frac{1}{4}-\varepsilon\right)\sqrt{m}$. We also give infinitely many graphs outside the exceptional family showing that no constant larger than $\frac{1}{4}$ is possible. Thus $\frac{1}{4}$ is the optimal asymptotic constant in the problem of Zhai et al.
Many extremal problems can be viewed as finding large independent sets in an auxiliary hypergraph. We propose a generalization of this by looking for ``large''independent sets $I$ in a hypergraph $\mathcal{F}$ where ``large''is measured by how many edges $I$ induces in another hypergraph $\mathcal{H}$ on the same vertex set as $\mathcal{F}$. We prove general supersaturation results for such extremal problems motivated by the breakthrough work of Ferber, McKinley and Samotij on counting $F$-free graphs. As applications, we prove new supersaturation bounds for generalized Tur\'an problems, as well as supersaturation bounds for a new set of extremal problems inspired by work of Fox and Pohoata on finding subsets $A\sub\mathbb{N}$ which maximize the number of solutions to a given system of equations while avoiding solutions to another system.
Sam Spiro· 0 citations
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