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学者姓名：林启忠
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The multicolor Ramsey number rk(C4) is the smallest integer N such that any k-edge coloring of KN contains a monochromatic C4. The current best upper bound of rk(C4) was obtained by Chung (1974) and independently by Irving (1974), i.e., rk(C4) <= k2 + k + 1 for all k >= 2. There is no progress on the upper bound since then. In this paper, we improve the upper bound of rk(C4) by showing that rk(C4) <= k2 + k - 1 for even k >= 6. The improvement is based on the upper bound of the Tur & aacute;n number ex(n, C4), in which we mainly use the double counting method and many novel ideas from Firke, Kosek, Nash, and Williford [J. Combin. Theory, Ser. B 103 (2013), 327-336].
Keyword ：
4-cycle 4-cycle Multicolor Ramsey number Multicolor Ramsey number Tur & aacute;n number Tur & aacute;n number
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| GB/T 7714 | Li, Tian-yu , Lin, Qi-zhong . Upper Bounds on the Multicolor Ramsey Numbers rk(C4) [J]. | ACTA MATHEMATICAE APPLICATAE SINICA-ENGLISH SERIES , 2025 , 41 (1) : 286-294 . |
| MLA | Li, Tian-yu 等. "Upper Bounds on the Multicolor Ramsey Numbers rk(C4)" . | ACTA MATHEMATICAE APPLICATAE SINICA-ENGLISH SERIES 41 . 1 (2025) : 286-294 . |
| APA | Li, Tian-yu , Lin, Qi-zhong . Upper Bounds on the Multicolor Ramsey Numbers rk(C4) . | ACTA MATHEMATICAE APPLICATAE SINICA-ENGLISH SERIES , 2025 , 41 (1) , 286-294 . |
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Let B-n be the book graph which consists of n copies of triangles all sharing a common edge. Let Cm be a cycle of length m. In 1978, Rousseau and Sheehan initiated the study of the book-cycle Ramsey number. A lot of effort has been made to determine the value of r(Bn,Cm) since then. In [Ars Combin., 31 (1991), pp. 239-248], Faudree, Rousseau, and Sheehan mentioned the following: "we know practically nothing about r(B-n,C-m) when m is even and greater than four. Also, the problem of computing r(Bn,C-m) when m is odd and m and n are nearly equal provides an unanswered test of strength." Answering the second part of the question above, the second and fifth authors recently obtained the value of r(B-n,C-m) for 8n/9+112 <= m <= inverted right perpendicular3n/2inverted left perpendicular+1 and n being large. However, the value of r(B-n,C-m) is previously unknown for m <= 8n/9+111 and m being even as well as n+13/4<m <= 8n/9+111 and m being odd. In this paper, for even m, we manage to determine the value of r(B-n,C-m) provided that m is linear with n and m is large enough. Thus this makes progress towards the first part of the question above. In addition, for odd m, we are able to obtain the value of r(B-n,C-m) for n/4 <= m <= 9/n/10 and m being large.
Keyword ：
book book cycle cycle pancyclic pancyclic Ramsey number Ramsey number
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| GB/T 7714 | Hu, Fu-tao , Lin, Qizhong , Luczak, Tomasz et al. RAMSEY NUMBERS OF BOOKS VERSUS LONG CYCLES [J]. | SIAM JOURNAL ON DISCRETE MATHEMATICS , 2025 , 39 (1) : 550-561 . |
| MLA | Hu, Fu-tao et al. "RAMSEY NUMBERS OF BOOKS VERSUS LONG CYCLES" . | SIAM JOURNAL ON DISCRETE MATHEMATICS 39 . 1 (2025) : 550-561 . |
| APA | Hu, Fu-tao , Lin, Qizhong , Luczak, Tomasz , Ning, Bo , Peng, Xing . RAMSEY NUMBERS OF BOOKS VERSUS LONG CYCLES . | SIAM JOURNAL ON DISCRETE MATHEMATICS , 2025 , 39 (1) , 550-561 . |
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For graphs H (1) and H (2) , the Ramsey number r(H-1, H (2) ) is the smallest positive integer N such that any graph G on N vertices contains H (1) as a subgraph, or its complement contains H (2) as a subgraph. Let B ((k)) (n) denote the book graph on n + k vertices which consists of n copies of K (k +1) all sharing a common K (k) , and let H := K-p(a(1), ..., a(p)) be the complete p-partite graph with parts of sizes a 1 ,... , a p . Recently, strengthening a result of Fox, He and Wigderson ( Adv. Combin. 4 (2023), 21pp), Fan and Lin (J. Combin. Theory Ser. A 199(2023), 19pp) showed that for every k, p, t >= 2, there exists delta > 0 such that the following holds for all large n . Let 1 <= a (1) <= center dot center dot center dot <= a (p -1) <= t and a(p) <= delta n be positive integers. If a (1) = 1, then r(H, B ((k)) (n) ) <= (p - 1 )( n + ka (2) - 1 ) + 1. The inequality is tight if n equivalent to 1 ( mod a 2 ) . In this paper, we improve the above upper bounds for the cases when n equivalent to 2 ( mod a 2 ) and n equivalent to 3 ( mod a (2) ) . Combining the new upper bounds and constructions of the lower bounds for these cases, we are able to determine the exact values of r ( K (p) ( a (1) ,... ,a(p)), B (n) ((k)) ) when p = 3. The bound on 1/delta we obtain is not of tower-type since our proof does not rely on the regularity lemma.
Keyword ：
Book Book Complete multipartite graph Complete multipartite graph Ramsey number Ramsey number
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| GB/T 7714 | Fan, Chunchao , Huang, Junqiang , Lin, Qizhong . Ramsey numbers of large books versus multipartite graphs [J]. | GRAPHS AND COMBINATORICS , 2024 , 40 (6) . |
| MLA | Fan, Chunchao et al. "Ramsey numbers of large books versus multipartite graphs" . | GRAPHS AND COMBINATORICS 40 . 6 (2024) . |
| APA | Fan, Chunchao , Huang, Junqiang , Lin, Qizhong . Ramsey numbers of large books versus multipartite graphs . | GRAPHS AND COMBINATORICS , 2024 , 40 (6) . |
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Given integers p, q >= 2, 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. Given a function f(n), the Ramsey-Turan 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 delta > 0, let rho(p, q, delta) = lim(n ->infinity) RT(n,p,q,delta n)/n(2). We always call rho(p, q) := lim(delta -> 0) rho(p, q, delta) the Ramsey-Turan density of K-p and K-q. In 1993, Erdos, Hajnal, Simonovits, Sos, and Szemeredi proposed to determine the value of rho(3, q) for q >= 3, and they conjectured that for q >= 2, rho(3, 2q - 1) = 1/2 (1 - 1/r(3,q)-1). More recently, in 2019, Kim, Kim, and Liu conjectured that for q >= 2, rho(3, 2q) = 1/2(1 - 1/r(3,q)). Erdos et al. (1993) determined rho(3, q) for q = 3, 4, 5 and rho(4, 4). There has been no progress on the Ramsey-Turan density rho(p, q) in the past 30 years. In this paper, we obtain rho(3, 6) = 5/12 and rho(3, 7) = 7/16. Moreover, we show that the corresponding asymptotically extremal structures are weakly stable, which answers a problem of Erdos et al. (1993) for the two cases.
Keyword ：
Ramsey graph Ramsey graph Ramsey-Turan number Ramsey-Turan number Szemeredi's regularity lemma Szemeredi's regularity lemma
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| GB/T 7714 | Hu, Xinyu , Lin, Qizhong . TWO-COLORED RAMSEY-TURAN DENSITIES INVOLVING TRIANGLES [J]. | SIAM JOURNAL ON DISCRETE MATHEMATICS , 2024 , 38 (3) : 2132-2162 . |
| MLA | Hu, Xinyu et al. "TWO-COLORED RAMSEY-TURAN DENSITIES INVOLVING TRIANGLES" . | SIAM JOURNAL ON DISCRETE MATHEMATICS 38 . 3 (2024) : 2132-2162 . |
| APA | Hu, Xinyu , Lin, Qizhong . TWO-COLORED RAMSEY-TURAN DENSITIES INVOLVING TRIANGLES . | SIAM JOURNAL ON DISCRETE MATHEMATICS , 2024 , 38 (3) , 2132-2162 . |
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For graphs $G$ and $H$ , the Ramsey number $r(G,H)$ is the smallest positive integer $N$ such that any red/blue edge colouring 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 $0\lt \alpha \lt 1$ , the random lower bound $r(B_{\lceil \alpha n\rceil },B_n)\ge (\sqrt{\alpha }+1)<^>2n+o(n)$ is not tight. In other words, there exists some constant $\beta \gt (\sqrt{\alpha }+1)<^>2$ such that $r(B_{\lceil \alpha n\rceil },B_n)\ge \beta n$ for all sufficiently large $n$ . This conjecture holds for every $\alpha \lt 1/6$ by a result of Nikiforov and Rousseau from 2005, which says that in this range $r(B_{\lceil \alpha n\rceil },B_n)=2n+3$ 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 $1/4\leq \alpha \leq 1$ . Moreover, we show that for any $1/6\leq \alpha \le 1/4$ and large $n$ , $r(B_{\lceil \alpha n\rceil }, B_n)\le \left (\frac 32+3\alpha \right ) n+o(n)$ , where the inequality is asymptotically tight when $\alpha =1/6$ or $1/4$ . We also give a lower bound of $r(B_{\lceil \alpha n\rceil }, B_n)$ for $1/6\le \alpha \lt \frac{52-16\sqrt{3}}{121}\approx 0.2007$ , showing that the random lower bound is not tight, i.e., the conjecture of Conlon, Fox, and Wigderson holds in this interval.
Keyword ：
Book Book Ramsey number Ramsey number refined regularity lemma refined regularity lemma
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| GB/T 7714 | Fan, Chunchao , Lin, Qizhong , Yan, Yuanhui . On a conjecture of Conlon, Fox, and Wigderson [J]. | COMBINATORICS PROBABILITY & COMPUTING , 2024 , 33 (4) : 432-445 . |
| MLA | Fan, Chunchao et al. "On a conjecture of Conlon, Fox, and Wigderson" . | COMBINATORICS PROBABILITY & COMPUTING 33 . 4 (2024) : 432-445 . |
| APA | Fan, Chunchao , Lin, Qizhong , Yan, Yuanhui . On a conjecture of Conlon, Fox, and Wigderson . | COMBINATORICS PROBABILITY & COMPUTING , 2024 , 33 (4) , 432-445 . |
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Given a graph F , let L ( F ) be a fixed finite family of graphs consisting of a C 4 and some bipartite graphs relying on an s-partite subgraph partitioning of edges of F . Define ( m , n, , a, , b) )-graph by an m x n bipartite graph with n >= m such that all vertices in the part of size n have degree a and all vertices in the part of size m have degree b >= a . In this paper, building upon the work of Janzer and Sudakov (2023+) + ) and combining with the idea of Conlon, Mattheus, Mubayi and Verstra & euml;te (2023+) + ) we obtain that for each s >= 2, if there exists an L(F)-free ( F )-free ( m , n, , a, , b) )-graph, then there exists an F-free graph H* with at least na - contains a copy of K s . As applications, we obtain some upper bounds of general Erdos-Rogers functions for some special graphs of F . Moreover, we obtain the multicolor Ramsey numbers r k + 1 ( C 5 ; t) ) =(t3k7+1) ( t 3 k 7 + 1 ) and r k + 1 ( C 7 ; t) ) =(tk4+1), ( t k 4 + 1 ) , which improve that by Xu and Ge (2022) [24]. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Keyword ：
Erdos-Rogers problem Erdos-Rogers problem Hypergraph container method Hypergraph container method Ramsey number Ramsey number Random block construction Random block construction
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| GB/T 7714 | Hu, Xinyu , Lin, Qizhong . Ramsey numbers and a general Erdos-Rogers function [J]. | DISCRETE MATHEMATICS , 2024 , 347 (12) . |
| MLA | Hu, Xinyu et al. "Ramsey numbers and a general Erdos-Rogers function" . | DISCRETE MATHEMATICS 347 . 12 (2024) . |
| APA | Hu, Xinyu , Lin, Qizhong . Ramsey numbers and a general Erdos-Rogers function . | DISCRETE MATHEMATICS , 2024 , 347 (12) . |
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A book B-n is a graph which consists of n triangles sharing a common edge. Rousseau and Sheehan (1978) conjectured that r(B-m, B-n) <= 2(m+ n+ 1)+ c some constant c > 0. Let m = left Perpendicular alpha n right Perpendicular where 0 < alpha = 1 is a real number. A result of Nikiforov and Rousseau [Random Structures Algorithms 27 (2005), 379-400] implies that this conjecture holds in a stronger form for 0 < alpha <= 1/6 and large n. We prove that r(B-m, B-n) <= (3/2 + 3 alpha + o(1))n, where 1/4 < alpha < 1/2. This confirms the conjecture in a stronger form for 1/6 <= alpha < 1/2 and large n. As a corollary, r(B (inverted left Perpendicular n4 inverted right Perpendicular), B-n) = (9/4 + o(1))n. (c) 2023 Elsevier Ltd. All rights reserved.
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| GB/T 7714 | Chen, Xun , Lin, Qizhong . New upper bounds for Ramsey numbers of books [J]. | EUROPEAN JOURNAL OF COMBINATORICS , 2024 , 115 . |
| MLA | Chen, Xun et al. "New upper bounds for Ramsey numbers of books" . | EUROPEAN JOURNAL OF COMBINATORICS 115 (2024) . |
| APA | Chen, Xun , Lin, Qizhong . New upper bounds for Ramsey numbers of books . | EUROPEAN JOURNAL OF COMBINATORICS , 2024 , 115 . |
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For graphs F, G and H, let F -> (G, H) signify that any red/blue edge coloring of F contains either a red G or a blue H. The Ramsey number r(G, H) is defined to be the smallest integer r such that Kr -> (G, H). Let B(k) n be the book graph which consists of n copies of Kk+1 all sharing a common Kk, and let G := Kp+1(a1, a2, ..., ap+1) be the complete (p + 1)partite graph with a1 = 1, a2 | (n - 1) and ai <= ai+1. In this paper, avoiding the use of Szemer & eacute;di's regularity lemma, we show that for any fixed p >= 1, k >= 2 and sufficiently large n, K p(n+a2k-1)+1 \ K1,n+a2-2 -> (G, Bn (k) ). This implies that the star-critical Ramsey number r & lowast;(G, Bn(k)) = (p - 1)(n + a2k -1) + a2(k -1) + 1. As a corollary, r & lowast;(G, B(k) n ) = (p - 1)(n + k - 1) + k for a1 = a2 = 1 and ai <= ai+1. This solves a problem proposed by Hao and Lin (2018) [11] in a stronger form. (c) 2024 Published by Elsevier B.V.
Keyword ：
Book Book Ramsey number Ramsey number Stability-supersaturation lemma Stability-supersaturation lemma Star-critical Ramsey number Star-critical Ramsey number
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| GB/T 7714 | Chen, Xun , Lin, Qizhong , Niu, Lin . Star-critical Ramsey numbers involving large books [J]. | DISCRETE MATHEMATICS , 2024 , 348 (2) . |
| MLA | Chen, Xun et al. "Star-critical Ramsey numbers involving large books" . | DISCRETE MATHEMATICS 348 . 2 (2024) . |
| APA | Chen, Xun , Lin, Qizhong , Niu, Lin . Star-critical Ramsey numbers involving large books . | DISCRETE MATHEMATICS , 2024 , 348 (2) . |
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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 KN contains either a red F or a blue H. Let Cn be a cycle of length n and Fn be a fan consisting of n triangles all sharing a common vertex. In this paper, we prove that for all sufficiently large n, � (2 + 2a + o(1))n if 1/2 < a < 1, R(C2 �an �, Fn) = (4a + o(1))n if a > 1.
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| GB/T 7714 | You, Chunlin , Lin, Qizhong . Ramsey Numbers of Large Even Cycles and Fans [J]. | ELECTRONIC JOURNAL OF COMBINATORICS , 2023 , 30 (3) . |
| MLA | You, Chunlin et al. "Ramsey Numbers of Large Even Cycles and Fans" . | ELECTRONIC JOURNAL OF COMBINATORICS 30 . 3 (2023) . |
| APA | You, Chunlin , Lin, Qizhong . Ramsey Numbers of Large Even Cycles and Fans . | ELECTRONIC JOURNAL OF COMBINATORICS , 2023 , 30 (3) . |
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A graph H = (W, E-H) is said to have bandwidth at most b if there exists a labeling of W as w(1), w(2),..., w(n) such that vertical bar i - j vertical bar <= b for every edge w(i)w(j) is an element of E-H. We say that H is a balanced (beta, Delta)-graph if it is a bipartite graph with bandwidth at most beta vertical bar W vertical bar and maximum degree at most Delta, and it also has a proper 2-coloring chi : W -> [2] such that parallel to chi(-1)(1)vertical bar - vertical bar chi(-1)(2)parallel to <= beta vertical bar chi(-1)(2)|. In this paper, we prove that for every gamma > 0 and every natural number Delta, there exists a constant beta > 0 such that for every balanced (beta, Delta)-graph H on n vertices we have R(H, H, C-n) <= (3 + gamma)n for all sufficiently large odd n. The upper bound is sharp for several classes of graphs. Let theta(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 >= 1, R(theta(n,t), theta(n,t), Cnt+lambda) = (3t + o(1))n, where lambda = 0 if nt is odd and lambda = 1 if nt is even. In particular, we have R(C-2n, C-2n, C2n+1) = (6 + o(1))n, which is a special case of a result of Figaj and Luczak (2018).
Keyword ：
Cycle Cycle Ramsey number Ramsey number Regularity Lemma Regularity Lemma Small bandwidth Small bandwidth
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| GB/T 7714 | You, Chunlin , Lin, Qizhong . Three-Color Ramsey Number of an Odd Cycle Versus Bipartite Graphs with Small Bandwidth [J]. | GRAPHS AND COMBINATORICS , 2023 , 39 (3) . |
| MLA | You, Chunlin et al. "Three-Color Ramsey Number of an Odd Cycle Versus Bipartite Graphs with Small Bandwidth" . | GRAPHS AND COMBINATORICS 39 . 3 (2023) . |
| APA | You, Chunlin , Lin, Qizhong . Three-Color Ramsey Number of an Odd Cycle Versus Bipartite Graphs with Small Bandwidth . | GRAPHS AND COMBINATORICS , 2023 , 39 (3) . |
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