Ramsey size linear and generalization

Ramsey size linear and generalization Eng Keat Hng ∗ Meng Ji † Ander Lamaison ‡ Jan uary 8, 2026 Abstract More than thirt y years ago, Erdős, F audree, Rousseau, and Sc help posed a fundamental question in extremal graph theory: What is the optimal constan t c k suc h that r ( C 2 k +1 , G ) ≤ c k m for an y graph G with m edges and no isolated v ertices? In this paper, w e mak e a significan t step to wards answ ering this question by pro ving that r ( C 2 k +1 , G ) ≤ (2 + o (1)) m + p, where p denotes the num b er of v ertices in G . This result provides the first improv emen t on the original op en problem. A dditionally , w e extend the work of Go ddard and Kleitman and indep enden tly Sidorenk o, who prov ed that r ( K 3 , G ) ≤ 2 m + 1 for an y graph G with m edges and no isolated v ertices. W e generalize their findings to the clique version, establishing that r ( K r , G ) ≤ c r m ( r − 1) / 2 , and to the m ulticolor setting, showing that r k +1 ( K 3 ; G ) ≤ c k m ( k +1) / 2 . Keyw ords: Ramsey num ber; Ramsey size linear; multicolor Ramsey n umber AMS sub ject classification 2020: 05C15; 05D10 1 In tro duction Giv en tw o graphs H and G , the R amsey numb er r ( H, G ) is the least p ositiv e integer N such that ev ery red-blue edge coloring of K N con tains a red cop y of H or a blue cop y of G . Ramsey theory is a fundamental area of com binatorics that studies the conditions under which order must app ear within c haos. Understanding and studying Ramsey n um b ers is one of the cen tral problems in combinatorics, and while m uc h progress has b een made in sp ecial cases, the general problem remains largely unsolv ed. W e refer the reader to the dynamic survey of Radziszo wski [12]. ∗ Extremal Combinatorics and Probabilit y Group (ECOPRO), Institute for Basic Science (IBS), Daejeon, South K orea. Supp orted by IBS-R029-C4. hng@ibs.re.kr † Corresp onding author: Sc ho ol of Mathematical Sciences, and Institute of Mathematics and In terdisciplinary Sciences, Tianjin Normal Universit y , Tianjin, China. Supp orted by the Tianjin Municipal Education Commission Scien tific Research Program Pro ject (Grant No. 2025KJ133) and the Institute of Basic Science (IBS-R029-C4). mji@tjnu.edu.cn ‡ Univ ersidad Pública de Na v arra, Pamplona, Spain. Supported b y the Institute of Basic Science (IBS-R029-C4). lamaison@mail.muni.cz 1 In the early 1980s, Harary conjectured that for an y graph G with m edges and no isolated v ertices, the Ramsey num ber r ( K 3 , G ) is at most 2 m + 1 . This conjecture was later pro ven b y Go ddard and Kleitman [9], and indep enden tly Sidorenko [13] in the 1990s. This result is tight, as equalit y is attained when G is a tree or a matc hing. Building on this, Erdős, F audree, Rousseau and Schelp [7] in tro duced the concept of Ramsey size linear in 1993. A graph H is said to b e R amsey size line ar if there is a constan t C such that for any graph G with m edges and no isolated v ertices, the Ramsey n um b er r ( H , G ) is b ounded ab o ve b y C · m . In other words, the Ramsey n um b er grows linearly with the n um b er of edges in G . The study of Ramsey size linear graphs has led to sev eral imp ortan t results. Erdős, F audree, Rousseau, and Sc help [7] prov ed that every graph H with p v ertices and q ≥ 2 p − 2 edges is not Ramsey size linear and ev ery connected graph H with p v ertices and q ≤ p + 1 edges is Ramsey size linear. F urthermore, they show ed that b oth b ounds on q are sharp. Several sp ecial classes of graphs w ere shown to be Ramsey size linear and b ounds on the Ramsey n um b ers were determined; see [1, 3, 7, 11]. Erdős, F audree, Rousseau and Sc help [7] studied the case that H is a cycle, and pro v ed that if k ≥ 2 and G is a c onne cte d gr aph of size m , then for m sufficiently lar ge r ( C 2 k , G ) ≤ m + 22 k √ m . Ho w ever, analogue upp er b ound has not been obtianed for the odd cycle, so they posed the follo wing question, whic h app ears as Problems 569 on Blo om’s Erdős problems website [2]. Question 1 ([2, 7]) . What is the b est p ossible c k such that r ( C 2 k +1 , G ) ≤ c k m for any gr aph G with m e dges without isolate d vertic es? Recall that the case k = 1 of Question 1 has b een solved by Go ddard and Kleitman [9], and Sidorenk o [13] indep enden tly . Theorem 1 ([9, 13]) . F or any gr aph G with m e dges and no isolate d vertic es, the R amsey numb er r ( K 3 , G ) is at most 2 m + 1 . As the main result of this pap er, w e obtain an upper bound on the Ramsey n um b er r ( C 2 k +1 , G ) . Theorem 2. F or every k ≥ 2 ther e is a c onstant B k such that for any gr aph G on p vertic es with m e dges and no isolate d vertic es we have r ( C 2 k +1 , G ) ≤ 2 m  1 + B k m − 1 / 20  + p . W e study tw o kinds of generalizations of Theorem 1, and first prov e the following clique v ersion. Theorem 3. F or every r ≥ 3 ther e is a c onstant c r such that for any gr aph G with m e dges and without isolate d vertic es we have r ( K r , G ) ≤ c r m r − 1 2 . 2 Secondly , we derive the m ulticolor setting of Theorem 1. Given graphs K 3 and G , the multi- c olor R amsey numb er r k +1 ( K 3 ; G ) is the least N such that a cop y of K N edge-colored with k + 1 colors contains a mono c hromatic K 3 in one of the first k colors or a mono c hromatic G in the last color. Theorem 4. F or every k ≥ 1 ther e is a c onstant c k such that for any gr aph G with m e dges and without isolate d vertic es we have r k +1 ( K 3 ; G ) ≤ c k m k +1 2 . Note that the Ramsey num b er on K 3 v ersus G is r 2 ( K 3 ; G ) = r ( K 3 , G ) . 2 Pro ofs In this section, w e giv e the pro ofs of Theorems 2, 3 and 4. Before pro ving the abov e three theorems, w e shall sho w a useful lemma, whose pro of needs the follo wing kno wn results. Theorem 5 ([4]) . F or every tr e e T on n vertic es and every p ositive inte ger p , we have r ( T , K p ) = ( n − 1)( p − 1) + 1 . Theorem 6 ([8]) . F or p ≥ k ≥ 3 every gr aph on p vertic es with at le ast ( k − 1)( p − 1) / 2 e dges c ontains a p ath of length k − 1 . No w we pro v e the lemma. Here we denote the path on n v ertices by P n . Lemma 1. F or every inte ger ℓ ≥ 4 , every r e al numb er β ≥ 3 and every gr aph G on p vertic es with m e dges and no isolate d vertic es, we have r ( P ℓ , G ) ≤  1 + 1 β  p + ( β + 1) 2 ℓ √ m . Pr o of. Let N =  1 + 1 β  p + ( β + 1) 2 ℓ √ m . T ak e a red/blue-colored copy K of K N and suppose that it con tains no red cop y of P ℓ . W riting d red ( v ; K ) for the red degree of v ∈ V ( K ) in K , let L = { u ∈ V ( K ) | d red ( u ; K ) ≥ ( β + 1) ℓ } . Note that | L | ≤ N β +1 . Indeed, supp ose not. Then the red subgraph of K has at least ℓN / 2 edges. Hence, it contains a red copy of P ℓ b y Theorem 6, giving a con tradiction. Let H b e obtained from K by deleting the v ertices in L . Note that | V ( H ) | ≥ β N β +1 and for all v ∈ V ( H ) we ha ve red degree d red ( v ; H ) ≤ ( β + 1) ℓ in H . W e shall show that H , and so K , contains a blue copy of G . First w e enumerate the vertices of G in decreasing degree order as v 1 , . . . , v p , that is, w e ha v e d G ( v 1 ) ≥ d G ( v 2 ) ≥ . . . ≥ d G ( v p ) . F or 1 ≤ r ≤ p let G r = G [ { v 1 , . . . , v r } ] . Now supp ose for a con tradiction that H con tains a blue cop y L of G r but not a blue cop y of G r +1 . W rite W for set of v ertices in L corresp onding to 3 the neighbors of v r +1 in G r +1 . Since H con tains no blue copy of G r +1 , w e immediately ha ve the follo wing observ ation. Each v ∈ V ( H ) \ V ( L ) is joine d to W by at le ast one r e d e dge. (A) T o finish the pro of, w e consider the following tw o cases. Case 1. 1 ≤ r ≤ p 2 . The graph H contains no red cop y of P ℓ , so b y Theorem 5 it con tains a blue cop y of K β N ( β +1) ℓ . In particular, we ha v e r ≥ β N ( β +1) ℓ . Now w e use a double coun ting argument on D = P w ∈ W d red ( w ; H ) to obtain a lo wer b ound on W . On the one hand, Observ ation A tells us that there are at least N − | L | − r ≥ p 2 + ( β + 1) β ℓ √ m red edges coming out from W , so we ha v e D ≥ p 2 + ( β + 1) β ℓ √ m . On the other hand, we hav e d red ( v ; H ) ≤ ( β + 1) ℓ for all v ∈ V ( H ) , so D ≤ | W | ( β + 1) ℓ . Hence, w e obtain | W | ≥ p 2 + ( β + 1) β ℓ √ m ( β + 1) ℓ ≥ β √ m . (1) W e shall also obtain an upp er b ound on | W | . Since the v ertices of G w ere en umerated in decreasing degree order, w e hav e | W | = d G ( v r +1 ) < 2 m r ≤ 2 m ( β + 1) ℓ β N ≤ 2 √ m β ( β + 1) . (2) Com bining (1) and (2) gives β 2 ( β + 1) < 2 , which con tradicts β ≥ 3 . Case 2. p 2 < r < p . In this case, w e ha v e | V ( H ) \ V ( L ) | > β ( β + 1) ℓ √ m . By Observ ation A and the pigeonhole principle, there is a v ertex v ∈ W with d red ( v ; H ) > β ( β +1) ℓ √ m | W | . Since the decreasing degree order of G giv es | W | ≤ 2 m/r and we hav e d red ( v ; H ) ≤ ( β + 1) ℓ , w e obtain ( β + 1) ℓ ≥ d red ( v ; H ) > β ( β + 1) ℓ √ m | W | > β ( β + 1) ℓp 4 √ m > β ( β + 1) ℓ 2 √ 2 , where the final inequality uses p > √ 2 m . But this giv es a contradiction b ecause β ≥ 3 . This completes the pro of of Lemma 1. 2.1 Pro of of Theorem 2 The following result will b e used in the pro of of Theorem 2. Theorem 7 ([6]) . F or al l t ≥ 3 and p ≥ 2 , writing ℓ =  t − 1 2  , we have r ( C t , K p ) ≤ ( t − 2) p 1+ 1 ℓ + (2 t − 3) p. 4 Equipp ed with these preliminary results, w e turn to the pro of of our main theorem. Let k ≥ 2 . Fix a sufficiently large constant B k ≥ 20 k 2 so that for every graph G on p vertices with m ≤ 3 20 edges and no isolated vertices, w e hav e r ( C 2 k +1 , G ) ≤ 2 m  1 + B k m − 1 / 20  + p . W e shall pro v e by induction on p that every graph G on p vertices with m edges and no isolated v ertices has r ( C 2 k +1 , G ) ≤ N := 2 m  1 + B k m − 1 / 20  + p . Note that b y our c hoice of B k this clearly holds whenever m ≤ 3 20 ; in particular, this cov ers all p ≤ 3 10 and gives a base case for the induction. Hence, w e assume that m > 3 20 . Without loss of generality , w e may also assume that G is connected b ecause r ( C 2 k +1 , G 1 ∪ G 2 ) ≤ max { r ( C 2 k +1 , G 1 ) + r ( C 2 k +1 , G 2 ) } holds for an y disjoin t union G 1 ∪ G 2 of graphs. W e pro ceed with the following tw o cases. Case 1. m ≥ (2 k − 1) p 1+max { 1 k , 1 19 } . By Theorem 7, w e ha v e r ( C 2 k +1 , G ) ≤ r ( C 2 k +1 , K p ) ≤ (2 k − 1) p 1+ 1 k + (4 k − 1) p ≤ N . Case 2. m < (2 k − 1) p 1+max { 1 k , 1 19 } . T ake a red/blue-colored copy K of K N and supp ose that it contains neither a red cop y of C 2 k +1 nor a blue copy of G . Obtain G ′ from G by deleting a minim um degree v ertex v . Since | V ( G ′ ) | = p − 1 , b y the induction hypothesis K contains a blue copy L of G ′ . Let X b e the set of v ertices of K not in L and Y b e the set of vertices in L correspon ding to the neigh b ors of v in G . F or u ∈ Y write R ( u ) for the set of v ertices in X joined to u b y a red edge. Note that each v ertex x ∈ X is joined b y a red edge to at least one vertex in Y . Indeed, any vertex x ∈ X with no red edge to Y could b e used to extend L to a blue copy of G , giving a contradiction. Hence, w e hav e X ⊆ [ u ∈ Y R ( u ) . (3) W e pro ve the follo wing claim. Claim 1. F or al l u ∈ Y we have | R ( u ) | < r ( P 2 k , G ) . Pr o of. Supp ose for a con tradiction that | R ( u ) | ≥ r ( P 2 k , G ) . Since K do es not contain a blue cop y of G , the low er b ound on | R ( u ) | implies that there is a red copy P of P 2 k on R ( u ) . But now u completes P to a red cop y of C 2 k +1 , giving a con tradiction. ■ By (3) and Claim 1, we hav e N = ( p − 1) + | X | ≤ ( p − 1) + X u ∈ Y | R ( u ) | < p + | Y | · r ( P 2 k , G ) . No w by applying Lemma 1 with β = m 1 / 20 ≥ 3 , we obtain N < p + | Y | p  1 + m − 1 / 20  + 2 | Y | k ( m 1 / 20 + 1) 2 √ m 5 Observ e that the case condition gives p − 1 <  2 k − 1 m  1 − max { 1 k +1 , 1 20 } and we hav e | Y | ≤ 2 m/p b ecause v is a minimum degree v ertex of G . By applying these observ ations, we obtain N < p + | Y | p  1 + m − 1 / 20  + 2 | Y | k ( m 1 / 20 + 1) 2 √ m ≤ p + 2 m  1 + m − 1 / 20 + 8 k m 3 / 5 p − 1  ≤ p + 2 m  1 + m − 1 / 20 + 16 k 2 m − 1 / 15  ≤ N , whic h gives the required con tradiction. This completes the pro of of Theorem 2. □ 2.2 Pro of of Theorem 3 W e pro ceed by double induction on r and p = | V ( G ) | . F or all r ≥ 3 let c r := 2 r − 1 . (4) F or the base case r = 3 , w e hav e r ( K 3 , G ) ≤ 2 m + 1 ≤ 3 m from a result of Sidorenko [13]. Now assume r > 3 and that the theorem holds for smaller v alues of r . F or the base case p = 2 , w e hav e r ( K r , K 2 ) = r ≤ c r . N o w further assume p > 2 and that the theorem holds for smaller v alu es of p . Since r ( K r ; G 1 ∪ G 2 ) ≤ r ( K r ; G 1 ) + r ( K r ; G 2 ) holds for an y disjoint union G 1 ∪ G 2 of graphs, w e ma y assume without loss of generalit y that G is connected. T ake a red/blue-colored copy K of K N on N ≥ c r m r − 1 2 v ertices. Supp ose it has neither a red cop y of K r nor a blue copy of G . Obtain G ′ from G b y deleting a minimum degree v ertex v . Since | V ( G ′ ) | = p − 1 , b y the induction hypothesis K contains a blue copy L of G ′ . Let X b e the set of v ertices of K not in L and Y b e the set of vertices in L correspon ding to the neigh b ors of v in G . F or u ∈ Y write R ( u ) for the set of v ertices in X joined to u b y a red edge. Note that each v ertex x ∈ X is joined b y a red edge to at least one v ertex in Y . Indeed, w e could use any vertex x ∈ X with no red edge to Y to extend L to a blue cop y of G , giving a con tradiction. Hence, we ha v e X ⊆ [ u ∈ Y R ( u ) . (5) W e pro ve the follo wing claim. Claim 2. F or al l u ∈ Y we have | R ( u ) | < r ( K r − 1 , G ) . Pr o of. Supp ose for a contradiction that | R ( u ) | ≥ r ( K r − 1 , G ) . Since K do es not contain a blue cop y of G , the lo w er b ound on | R ( u ) | implies that there is a red edge on R ( u ) . But this together with u giv es a red cop y of K 3 , which gives a contradiction. ■ By (5) and Claim 2, we obtain N = ( p − 1) + | X | ≤ ( p − 1) + X u ∈ Y | R ( u ) | < p + | Y | · r ( K r − 1 , G ) . 6 Since v is a minimum degree v ertex of G , we ha v e | Y | ≤ √ 2 m . By applying this together with the b ound on r ( K r − 1 , G ) from the inductive h yp othesis, p ≤ m + 1 and (4), we get N < p + | Y | · r ( K r − 1 , G ) ≤ p + √ 2 m · c r − 1 m r 2 − 1 ≤ c r m r − 1 2 ≤ N , whic h gives the required con tradiction. □ 2.3 Pro of of Theorem 4 W e pro ceed by double induction on k and p = | V ( G ) | ≥ 2 . F or all k ≥ 1 let c k := 3 · 2 k − 1 · k ! . (6) F or the case k = 1 , w e ha ve r 2 ( K 3 ; G ) ≤ 2 m + 1 ≤ c 1 m from a result of Sidorenko [13]. Now assume k > 1 and that the theorem holds for smaller k . F or the case p = 2 , w e ha v e r k +1 ( K 3 ; K 2 ) = r k ( K 3 ; K 3 ) ≤ 3 · k ! ≤ c k from a result of Greenw oo d and Gleason [10]. W e further assume p > 2 and that the theorem holds for smaller p . Without loss of generality , w e ma y assume that G is connected b ecause r k +1 ( K 3 ; G 1 ∪ G 2 ) ≤ r k +1 ( K 3 ; G 1 ) + r k +1 ( K 3 ; G 2 ) holds for any disjoint union G 1 ∪ G 2 of graphs. F or N ≥ c k m k +1 2 tak e a copy K of K N edge-colored with colors 1 , . . . , k + 1 . Supp osethat it con tains neither a mono chromatic cop y of K 3 in some color i ∈ { 1 , . . . , k } nor a copy of G in color k + 1 . Let G ′ b e obtained from G b y deleting a minimum degree v ertex v . Since | V ( G ′ ) | = p − 1 , b y the induction h yp othesis K contains a copy L of G ′ in color k + 1 . Let X b e the set of v ertices of K not in L and Y be the set of v ertices in L corresp onding to the neighbors of v in G . F or 1 ≤ i ≤ k and u ∈ Y write W i ( u ) for the set of v ertices in X joined to u by an edge in color i . Note that eac h v ertex x ∈ X is joined to at least one vertex in Y b y an edge not in color k + 1 . Indeed, we could use any vertex x ∈ X for which this do es not hold to extend L to a copy of G in color k + 1 , giving a contradiction. Hence, w e hav e X ⊆ [ u ∈ Y   [ i ∈{ 1 ,...,k } W i ( u )   . (7) W e ha ve the follo wing claim. Claim 3. F or al l i ∈ { 1 , . . . , k } and u ∈ Y we have | W i ( u ) | < r k ( K 3 ; G ) . Pr o of. Supp ose for a contradiction that | W i ( u ) | ≥ r k ( K 3 ; G ) . Since K contains neither a copy of G in color k + 1 nor a mono c hromatic copy of K 3 in some color j ∈ { 1 , . . . , k } \ { i } , the lo w er b ound on | W i ( u ) | implies that there is an edge on W i ( u ) with color i . But this together with u giv es a cop y of K 3 in color i , whic h giv es a con tradiction. ■ By (7) and Claim 3, we obtain N = ( p − 1) + | X | ≤ ( p − 1) + X u ∈ Y   X i ∈{ 1 ,...,k } | W i ( u ) |   < p + | Y | · k · r k ( K 3 ; G ) . 7 Since v is a minimum degree v ertex of G , we ha v e | Y | ≤ √ 2 m . By applying this together with the b ound on r k ( K 3 ; G ) from the inductive h yp othesis, p ≤ m + 1 and (6), we get N < p + | Y | · k · r k ( K 3 ; G ) ≤ p + √ 2 m · k · c k − 1 m k 2 ≤ c k m k +1 2 ≤ N , whic h gives the required con tradiction. □ 3 Concluding remarks It is worth men tioning that Erdős, F audree, Rousseau and Schelp ask ed a m uc h more difficult and sp ecific problem which is to extend the result of Sidorenk o on triangles to cycles of arbitrary length in the follo wing, which was confirmed if k = 3 , 4 , 5 in [12]. Question 2 ([2, 7]) . Is r ( C k , G ) ≤ 2 m + ⌊ k − 1 2 ⌋ , wher e k ≥ 3 and G is a gr aph of size m without isolate d vertic es? They also ask ed the follo wing question in [2, 5, 7]. Question 3 ([2, 5, 7]) . Is K 3 , 3 R amsey size line ar? A c kno wledgemen t. The authors gratefully ac knowledge the many helpful suggestions and dis- cussions of Professor Hong Liu during the preparation of the pap er. The second author also wishes to thank the ECOPR O group for their w arm hospitalit y . The authors also wish to express their gratitude to Y ulai Ma for his careful reading of the man uscript. Declaration of Comp eting Interests. The authors declare that they hav e no known competing financial in terests or p ersonal relationships that could ha ve app eared to influence the w ork rep orted in this pap er. References [1] P . N. Balister, R. H. Schelp, and M. Simonovits. A note on Ramsey size-linear graphs. J. Gr aph The ory , 39(1):1–5, 2002. [2] T. F. Blo om. Erdős Problems # 569, https://www.erdosproblems.com/569. Accessed F ebru- ary 28, 2025. [3] D. Bradač, L. Gish b oliner, and B. Sudako v. On Ramsey size-linear graphs and related ques- tions. SIAM J. Discr ete Math. , 38(1):225–242, 2024. [4] V. Chv átal. T ree-complete graph Ramsey n um b ers. J. Gr aph The ory , 1(1):93, 1977. [5] P . Erdős. Some of m y fav ourite problems in n um b er theory , combinatorics, and geometry . 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