arXiv subjects

Qizhong Lin

Publications and source records attributed to Qizhong Lin.

34 records · Page 2Linked to original sources

arXiv2025

Phase transitions of the Erdős-Gyárfás function

Given positive integers $p,q$. For any integer $k\ge2$, an edge coloring of the complete $k$-graph $K_n^{(k)}$ is said to be a $(p,q)$-coloring if every copy of $K_p^{(k)}$ receives at least $q$ colors. The Erdős-Gyárfás function $f_k(n,p,q)$ is the minimum number of colors that are needed for $K_n^{(k)}$ to have a $(p,q)$-coloring. Conlon, Fox, Lee and Sudakov (\emph{IMRN, 2015}) conjectured that for any positive integers $p, k$ and $i$ with $k\ge3$ and $1\le i<k$, $f_k(n,p,{{p-i}\choose{k-i}})=(\log_{(i-1)}n)^{o(1)}$, where $\log_{(i)}n$ is an iterated $i$-fold logarithm in $n$. It has been verified to be true for $k=3, p=4, i=1$ by Conlon et. al (\emph{IMRN, 2015}), for $k=3, p=5, i=2$ by Mubayi (\emph{JGT, 2016}), and for all $k\ge 4, p=k+1,i=1$ by B. Janzer and O. Janzer (\emph{JCTB, 2024}). In this paper, we give new constructions and show that this conjecture holds for infinitely many new cases, i.e., it holds for all $k\ge4$, $p=k+2$ and $i=k-1$.

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arXiv2025

Fan-complete Ramsey numbers

For graphs $G$ and $H$, we consider Ramsey numbers $r(G,H)$ with tight lower bounds, namely, $r(G,H) \geq (χ(G)-1)(|H|-1)+1,$ where $χ(G)$ denotes the chromatic number of $G$ and $|H|$ denotes the number of vertices in $H$. We say $H$ is $G$-good if the equality holds. Let $G+H$ be the join graph obtained from graphs $G$ and $H$ by adding all edges between the disjoint vertex sets of $G$ and $H$. Let $nH$ denote the union graph of $n$ disjoint copies of $H$. We show that $K_1+nH$ is $K_p$-good if $n$ is sufficiently large. In particular, the fan-graph $F_n=K_1 + n K_2$ is $K_p$-good if $n\geq 27p^2$, improving previous tower-type lower bounds for $n$ due to Li and Rousseau (1996). Moreover, we give a stronger lower bound inequality for Ramsey number $r(G, K_1+F)$ for the case of $G=K_p(a_1, a_2, \dots, a_p)$, the complete $p$-partite graph with $a_1=1$ and $a_i \leq a_{i+1}$. In particular, using a stability-supersaturation lemma by Fox, He and Wigderson (2021), we show that for any fixed graph $H$, \begin{align*} r(G,K_1+nH) = \left\{ \begin{array}{ll} (p-1)(n |H|+a_2-1)+1 & \textrm{if $n|H|+a_2-1$ is even or $a_2-1$ is even,}\\ (p-1)(n |H|+a_2-2)+1 & \textrm{otherwise,} \end{array} \right. \end{align*} where $G=K_p(1,a_2, \dots, a_p)$ with $a_i$'s satisfying some mild conditions and $n$ is sufficiently large. The special case of $H=K_1$ gives an answer to Burr's question (1981) about the discrepancy of $r(G, K_{1,n})$ from $G$-goodness for sufficiently large $n$. All bounds of $n$ we obtain are not of tower-types.

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arXiv2024

New bounds of two hypergraph Ramsey problems

We focus on two hypergraph Ramsey problems. First, we consider the Erdős-Hajnal function $r_k(k+1,t;n)$. In 1972, Erdős and Hajnal conjectured that the tower growth rate of $r_k(k+1,t;n)$ is $t-1$ for each $2\le t\le k$. To finish this conjecture, it remains to show that the tower growth rate of $r_4(5,4;n)$ is three. We prove a superexponential lower bound for $r_4(5,4;n)$, which improves the previous best lower bound $r_4(5,4;n)\geq 2^{Ω(n^2)}$ from Mubayi and Suk (\emph{J. Eur. Math. Soc., 2020}). Second, we prove an upper bound for the hypergraph Erdős-Rogers function $f^{(k)}_{k+1,k+2}(N)$ that is an iterated $(k-3)$-fold logarithm in $N$ for each $k\geq 5$. This improves the previous upper bound that is an iterated $(k-13)$-fold logarithm in $N$ for $k\ge14$ due to Mubayi and Suk (\emph{J. London Math. Soc., 2018}), in which they conjectured that $f^{(k)}_{k+1,k+2}(N)$ is an iterated $(k-2)$-fold logarithm in $N$ for each $k\ge3$.

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arXiv2024

Sharp Ramsey thresholds for large books

For graphs $G$ and $H$, let $G\to H$ signify that any red/blue edge coloring of $G$ contains a monochromatic $H$. Let $G(N,p)$ be the random graph of order $N$ and edge probability $p$. The Ramsey thresholds for fixed graphs have received most attention. In this paper, we consider the Ramsey thresholds in another angle. In particular, we will consider the sharp Ramsey threshold for the large book graph $B_n^{(k)}$, which consists of $n$ copies of $K_{k+1}$ all sharing a common $K_k$. In particular, for every fixed integer $k\ge 2$ and for any real $c>1$, let $N=c2^k n$. Then for any real $γ>0$, \[ \lim_{n\to \infty} \Pr(G(N,p)\to B_n^{(k)})= \left\{ \begin{array}{cl} 0 & \mbox{if $p\le\frac{1}{c^{1/k}}(1-γ)$,} \\ 1 & \mbox{if $p\ge\frac{1}{c^{1/k}}(1+γ)$}. \end{array} \right. \] This implies that $r(B_n^{(k)},B_n^{(k)})=2^kn+o(n)$, and hence especially extends the work of Conlon (2019) and the follow-up work of Conlon, Fox and Wigderson (2022) on book Ramsey numbers.

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arXiv2024

On a conjecture of Conlon, Fox and Wigderson

For graphs $G$ and $H$, the Ramsey number $r(G,H)$ is the smallest positive integer $N$ such that any red/blue edge coloring of the complete graph $K_N$ contains either a red $G$ or a blue $H$. A book $B_n$ is a graph consisting of $n$ triangles all sharing a common edge. Recently, Conlon, Fox and Wigderson conjectured that for any $0<α<1$, the random lower bound $r(B_{\lceilαn\rceil},B_n)\ge (\sqrtα+1)^2n+o(n)$ is not tight. In other words, there exists some constant $β>(\sqrtα+1)^2$ such that $r(B_{\lceilαn\rceil},B_n)\ge βn$ for all sufficiently large $n$. This conjecture holds for every $α< 1/6$ by a result of Nikiforov and Rousseau from 2005, which says that in this range $r(B_{\lceilαn\rceil},B_n)=2n+3$ for all sufficiently large $n$. We disprove the conjecture of Conlon, Fox and Wigderson. Indeed, we show that the random lower bound is asymptotically tight for every $1/4\leq α\leq 1$. Moreover, we show that for any $1/6\leq α\le 1/4$ and large $n$, $r(B_{\lceilαn\rceil}, B_n)\le\left(\frac 32+3α\right) n+o(n)$, where the inequality is asymptotically tight when $α=1/6$ or $1/4$. We also give a lower bound of $r(B_{\lceilαn\rceil}, B_n)$ for $1/6\leα< \frac{52-16\sqrt{3}}{121}\approx0.2007$, showing that the random lower bound is not tight, i.e., the conjecture of Conlon, Fox and Wigderson holds in this interval.

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arXiv2023

Two Ramsey-Turán numbers involving triangles

Given integers $p, q\ge2$, we say that a graph $G$ is $(K_p,K_q)$-free if there exists a red/blue edge coloring of $G$ such that it contains neither a red $K_p$ nor a blue $K_q$. Fix a function $f( n )$, the Ramsey-Turán number $RT( {n,p,q,f( n ))} $ is the maximum number of edges in an $n$-vertex $(K_p,K_q)$-free graph with independence number at most $f( n )$. For any $δ>0$, let $ρ(p, q,δ) = \mathop {\lim }\limits_{n \to \infty } \frac{RT(n,p, q,δn)}{n^2}$. We always call $ρ(p, q):= \mathop {\lim }\limits_{δ\to 0}ρ(p, q,δ)$ the Ramsey-Turán density of $K_p$ and $K_q$. In 1993, Erdős, Hajnal, Simonovits, Sós and Szemerédi proposed to determine the value of $ρ(3,q)$ for $q\ge3$, and they conjectured that for $q \ge 2$, $ρ\left( {3,2q - 1} \right) = \frac{1}{2}(1 - \frac{1}{r(3,q) - 1})$. Recently, Kim, Kim and Liu (2019) conjectured that for $q \ge 2$, $ρ( {3,2q } ) = \frac{1}{2}( 1 - \frac{1}{r( {3,q} )})$. Erdős et al. (1993) determined $ρ(3,q)$ for $q=3,4,5$ and $ρ(4,4)$. There is no progress on the Ramsey-Turán density $ρ(p, q)$ in the past thirty years. In this paper, we obtain $ρ(3,6)=\frac{5}{12}$ and $ρ(3,7)=\frac{7}{16}$. Moreover, we show that the corresponding asymptotically extremal structures are weakly stable, which answers a problem of Erdős et al. (1993) for the two cases.

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arXiv2022

Ramsey numbers of large even cycles and fans

For graphs $F$ and $H$, the Ramsey number $R(F, H)$ is the smallest positive integer $N$ such that any red/blue edge coloring of $K_N$ contains either a red $F$ or a blue $H$. Let $C_n$ be a cycle of length $n$ and $F_n$ be a fan consisting of $n$ triangles all sharing a common vertex. In this paper, we prove that for all sufficiently large $n$, \[ R(C_{2\lfloor an\rfloor}, F_n)= \left\{ \begin{array}{ll} (2+2a+o(1))n & \textrm{if $1/2\leq a< 1$,}\\ (4a+o(1))n & \textrm{if $ a\geq 1$.} \end{array} \right. \]

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arXiv2022

Ramsey non-goodness involving books

In 1983, Burr and Erdős initiated the study of Ramsey goodness problems.Nikiforov and Rousseau (2009) resolved almost all goodness questions raised by Burr and Erdős, in which the bounds on the parameters are of tower type since their proofs rely on the regularity lemma. Let $B_{k,n}$ be the book graph on $n$ vertices which consists of $n-k$ copies of $K_{k+1}$ all sharing a common $K_k$, and let $H=K_p(a_1,\dots,a_{p})$ be the complete $p$-partite graph with parts of sizes $a_1,\dots,a_{p}$. Recently, avoiding use of the regularity lemma, Fox, He and Wigderson (2021) revisit several Ramsey goodness results involving books. They comment that it would be very interesting to see how far one can push these ideas. In particular, they conjecture that for all integers $k, p, t\ge 2$, there exists some $δ>0$ such that for all $n\ge 1$, $1\leq a_1\le\cdots\le a_{p-1}\le t$ and $a_p \le δn$, we have $r(H, B_{k,n})= (p-1)(n-1)+d_k(n,K_{a_1,a_2})+1,$ where $d_k(n,K_{a_1,a_2})$ is the maximum $d$ for which there is an $(n+d-1)$-vertex $K_{a_1,a_2}$-free graph in which at most $k-1$ vertices have degree less than $d$.They verify the conjecture when $a_1=a_2=1$. Building upon the work of Fox et al. (2021), we make a substantial step by showing that the conjecture "roughly" holds if $a_1=1$ and $a_2|(n-1-k)$, i.e. $a_2$ divides $n-1-k$. Moreover, avoiding use of the regularity lemma, we prove that for every $k, a\geq 1$ and $p\ge2$, there exists $δ>0$ such that for all large $n$ and $b\le δ\ln n$, $r(K_p(1,a,b,\dots,b), B_{k,n})= (p-1)(n-1)+k(p-1)(a-1)+1$ if $a|(n-1-k)$, where the case when $a=1$ has been proved by Nikiforov and Rousseau (2009) using the regularity lemma. The bounds on $1/δ$ we obtain are not of tower-type since our proofs do not rely on the regularity lemma.

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arXiv2022

Three-color Ramsey number of an odd cycle versus bipartite graphs with small bandwidth

A graph $\mathcal{H}=(W,E_\mathcal{H})$ is said to have {\em bandwidth} at most $b$ if there exists a labeling of $W$ as $w_1,w_2,\dots,w_n$ such that $|i-j|\leq b$ for every edge $w_iw_j\in E_\mathcal{H}$. We say that $\mathcal{H}$ is a {\em balanced $(β,Δ)$-graph} if it is a bipartite graph with bandwidth at most $β|W|$ and maximum degree at most $Δ$, and it also has a proper 2-coloring $χ:W\rightarrow[2]$ such that $||χ^{-1}(1)|-|χ^{-1}(2)||\leqβ|χ^{-1}(2)|$. In this paper, we prove that for every $γ>0$ and every natural number $Δ$, there exists a constant $β>0$ such that for every balanced $(β,Δ)$-graph $\mathcal{H}$ on $n$ vertices we have $$R(\mathcal{H}, \mathcal{H}, C_n) \leq (3+γ)n$$ for all sufficiently large odd $n$. The upper bound is sharp for several classes of graphs. Let $θ_{n,t}$ be the graph consisting of $t$ internally disjoint paths of length $n$ all sharing the same endpoints. As a corollary, for each fixed $t\geq 1$, $R(θ_{n, t},θ_{n, t}, C_{nt+λ})=(3t+o(1))n,$ where $λ=0$ if $nt$ is odd and $λ=1$ if $nt$ is even. In particular, we have $R(C_{2n},C_{2n}, C_{2n+1})=(6+o(1))n$, which is a special case of a result of Figaj and Łuczak (2018).

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arXiv2021

Ramsey numbers of large books

A book $B_n$ is a graph which consists of $n$ triangles sharing a common edge. In 1978, Rousseau and Sheehan conjectured that the Ramsey number satisfies $r(B_m,B_n)\le 2(m+n)+c$ for some constant $c>0$. In this paper, we obtain that $r(B_m, B_n)\le 2(m+n)+o(n)$ for all $m\le n$ and $n$ large, which confirms the conjecture of Rousseau and Sheehan asymptotically. As a corollary, our result implies that a related conjecture of Faudree, Rousseau and Sheehan (1982) on strongly regular graph holds asymptotically.

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arXiv2021

Ramsey numbers of quadrilateral versus books

A book $B_n$ is a graph which consists of $n$ triangles sharing a common edge. In this paper, we study Ramsey numbers of quadrilateral versus books. Previous results give the exact value of $r(C_4,B_n)$ for $1\le n\le 14$. We aim to show the exact value of $r(C_4,B_n)$ for infinitely many $n$. To achieve this, we first prove that $r(C_4,B_{(m-1)^2+(t-2)})\le m^2+t$ for $m\ge4$ and $0 \leq t \leq m-1$. This improves upon a result by Faudree, Rousseau and Sheehan (1978) which states that \begin{align*} r(C_4,B_n)\le g(g(n)), \;\;\text{where}\;\;g(n)=n+\lfloor\sqrt{n-1}\rfloor+2. \end{align*} Combining the new upper bound and constructions of $C_4$-free graphs, we are able to determine the exact value of $r(C_4,B_n)$ for infinitely many $n$. As a special case, we show $r(C_4,B_{q^2-q-2}) = q^2+q-1$ for all prime power $q\ge4$.

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arXiv2021

Large book--cycle Ramsey numbers

Let $B_n^{(k)}$ be the book graph which consists of $n$ copies of $K_{k+1}$ all sharing a common $K_k$, and let $C_m$ be a cycle of length $m$. In this paper, we first determine the exact value of $r(B_n^{(2)}, C_m)$ for $\frac{8}{9}n+112\le m\le \lceil\frac{3n}{2}\rceil+1$ and $n \geq 1000$. This answers a question of Faudree, Rousseau and Sheehan (Cycle--book Ramsey numbers, {\it Ars Combin.,} {\bf 31} (1991), 239--248) in a stronger form when $m$ and $n$ are large. Building upon this exact result, we are able to determine the asymptotic value of $r(B_n^{(k)}, C_n)$ for each $k \geq 3$. Namely, we prove that for each $k \geq 3$, $r(B_n^{(k)}, C_n)= (k+1+o_k(1))n.$ This extends a result due to Rousseau and Sheehan (A class of Ramsey problems involving trees, {\it J.~London Math.~Soc.,} {\bf 18} (1978), 392--396).

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arXiv2018

New upper bound for multicolor Ramsey number of odd cycles

Let $r_k(C_{2m+1})$ be the $k$-color Ramsey number of an odd cycle $C_{2m+1}$ of length $2m+1$. It is shown that for each fixed $m\ge2$, \[r_k(C_{2m+1}) 0$ is a constant. This improves an old result by Bondy and Erdős (Ramsey numbers for cycles in graphs, J. Combin. Theory Ser. B 14 (1973) 46-54).

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arXiv2018

Cycle Ramsey numbers for random graphs

Let $C_{n}$ be a cycle of length $n$. As an application of Szemerédi's regularity lemma, Łuczak ($R(C_n,C_n,C_n)\leq (4+o(1))n$, J. Combin. Theory Ser. B, 75 (1999), 174--187) in fact established that $K_{(8+o(1))n}\to(C_{2n+1},C_{2n+1},C_{2n+1})$. In this paper, we strengthen several results involving cycles. Let $\mathcal{G}(n,p)$ be the random graph. We prove that for fixed $0 0$, there exists an integer $n_0$ such that for all integer $n_3>n_0$, we have a.a.s. that \begin{align*} \mathcal{G}((8+δ)n_1,p) \to (C_{2n_1+1},C_{2n_2+1},C_{2n_3+1}). \end{align*} Moreover, we prove that for fixed $0 0$ with same order, i.e. $n_2=Θ(n_1)$ and $n_3=Θ(n_1)$, we have a.a.s. that \begin{align*} \mathcal{G}(2n_1+n_2+n_3+o(1)n_1,p) \to (C_{2n_1},C_{2n_2},C_{2n_3}). \end{align*} Similar results for the two color case are also obtained.

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arXiv2015

Sparse Multipartite Graphs as Partition Universal for Graphs of Bounded-Degrees

For graphs $G$ and $H$, let $G\to (H,H)$ signify that any red/blue edge coloring of $G$ contains a monochromatic $H$ as a subgraph, and $\mathcal{H}(Δ,n)=\{H:|V(H)|=n,Δ(H)\le Δ\}$. For fixed $Δ$ and $n$, we say that $G$ is a partition universal graph for $\mathcal{H}(Δ,n)$ if $G\to (H,H)$ for every $H\in\mathcal{H}(Δ,n)$. In 1983, Chvátal, Rödl, Szemerédi and Trotter proved that for any $Δ\ge2$ there exists a constant $B$ such that, for any $n$, if $N\ge Bn$ then $K_N$ is partition universal for $\mathcal{H}(Δ,n)$. Recently, Kohayakawa, Rödl, Schacht and Szemerédi proved that the complete graph $K_N$ in above result can be replaced by sparse graphs. They obtained that for fixed $Δ\ge2$, there exist constants $B$ and $C$ such that if $N\ge Bn$ and $p=C(\log N/N)^{1/Δ}$, then {\bf a.a.s.} $G(N,p)$ is partition universal graph for $\mathcal{H}(Δ,n)$, where $G(N,p)$ is the standard random graph on $N$ vertices with $\mathbb{P}(e)=p$ for each edge $e$. From some results of Bollobás and Łuczak, we know that {\bf a.a.s.} $χ(G(N,p)) = Θ((N/\log N)^{1-1/Δ})$. In this paper, we shall show that the $G(N,p)$ in above result can be replaced by random multipartite graph. Let $K_{r}(N)$ be the complete $r$-partite graph with $N$ vertices in each part, and $G_r(N,p)$ the random spanning subgraph of $K_r(N)$, in which each edge appears with probability $p$. It is shown that for fixed $Δ\ge2$ there exist constants $r, B$ and $C$ depending only on $Δ$ such that if $N\ge Bn$ and $p=C(\log N/N)^{1/Δ}$, then {\bf a.a.s.} $G_r(N,p)$ is partition universal graph for $\mathcal{H}(Δ,n)$. The proof mainly uses the sparse multipartite regularity lemma.

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arXiv2014

A Folkman Linear Family

For graphs $F$ and $G$, let $F\to (G,G)$ signify that any red/blue edge coloring of $F$ contains a monochromatic $G$. Define Folkman number $f(G;p)$ to be the smallest order of a graph $F$ such that $F\to (G,G)$ and $ω(F) \le p$. It is shown that $f(G;p)\le cn$ for graphs $G$ of order $n$ with $Δ(G)\le Δ$, where $Δ\ge 3$, $c=c(Δ)$ and $p=p(Δ)$ are positive constants.

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