Symmetric Sum-Free Partitions and Lower Bounds for Schur Numbers

Symmetric Sum-Free Partitions and Lower Bounds for Schur NumbersHarold Fredricksen Naval Postgraduate School, Department of Mathematics, Monterey,CA93943,USA halF@Melvin M.SweetInstitute for Defense Analyses, Center for Communications Research, 4320Westerra Court,San Diego,CA92121,USAsweet@Submitted:May9,2000;Accepted:May18,2000AbstractWe give new lower bounds for the Schur numbers S(6)and S(7).This will imply new lower bounds for the Multicolor Ramsey Numbers R6(3)and R7(3).We also make several observations concerning symmetric sum-free partitions into5sets.IntroductionA set of integers is said to be sum-free if for all i and j≥i in the set the sum i+j is not in the set.The Schur number S(k)is defined to be the max-imum integer n for which the interval[1,n]can be partitioned into k sum-free sets.1I.Schur[7]proved that S(k)isfinite,and thatS(k)≥3S(k−1)+1.(1) In the framework of Ramsey theory,Schur’s proof yields the inequalityS(k)≤R k(3)−2,(2) where R k(n)denotes the k-color Ramsey number,defined to be the smallest integer such that any k-color edge coloring of the complete graph on R k(n) vertices has at least one complete subgraph all of whose edges have the same color.the electronic journal of combinatorics7(2000),#R322 It is easily verified that S(1)=1,S(2)=4,and S(3)=13;and L.D. Baumert[1]showed in1961with the aid of a computer that S(4)=44.The best known bounds for S(5)are160≤S(5)≤315(3) Thefirst inequality in(3)is due to G.Exoo[4],and the second follows from(2) and the bounds for R5(3)given in S.Radziszowski’s survey paper[6].For earlier work on lower bounds for S(k)see H.Fredricksen[5]and A.Beutelspacher and W.Brestovansky[2].The best previously known lower bound S(6)≥481for S(6)follows from (1).At the end of this note we list constructions that showS(6)≥536S(7)≥1680.Using(2)we obtain the following lower bounds for the Ramsey numbers R6(3) and R7(3):R6(3)≥538R7(3)≥1682.This improves the bounds given in Radziszowski’s survey paper.Note that the bound for R7(3)is better than the lower bound of1662that can be obtained by using F.Chung’s[3]inequalityR k+1(3)≥3R k(3)+R k−2(3)−3,for k≥3,with our new bound for R6(3).The constructions and results in this paper focus on symmetric partitions, where a sum-free partition P of the interval[1,n]is said to be symmetric if all symmetric pairs i and(n+1)−i are in the same set;except in the case when n+1is divisible by3,we must allow(n+1)/3and2(n+1)/3to be in different sets.There are no real theorems in this note,only observations and conjectures based on a computer study of symmetric sum-free partitions.We have restricted our focus to such partitions because the search space is smaller, and the concepts of equivalence and depth which are explained below provide additional useful structure for restricting the search.It is easy to see that if a symmetric sum-free partition P is multiplied mod(n+1)by an integer m that is relatively prime to n+1,then m P is also a symmetric sum-free partition.We will say two such partitions are equivalent.Given a partition P of[1,n],we define the depth,denoted d(P),to be the largest of the set minimums;and for a symmetric sum-free partition we define the e-depth,denoted D(P),to be the maximum of d(P )over all partitions P equivalent to P.We define D k(n)to be the maximum of D(P)over all symmetric sum-free partitions P of[1,n]into k sets.Note that we must haveD k(n)≤S(k−1)+1.the electronic journal of combinatorics7(2000),#R323 In a later section we will give exact values of D5(n)for many values of n. These values and other studies described later for partitions into5sets have led us to make the following conjectures:Conjecture1The largest integer for which there is a symmetric sum-free par-tition into5sets is160;and perhaps S(5)=160.Conjecture2All symmetric sum-free partitions of160into5sets have e-depth 44.Conjecture3There are no symmetric sum-free partitions of155or158into 5sets.Note155and158are the special cases in the definition of symmetric,and that it is not obvious that a symmetric partition of n into k sets implies that there is a symmetric partition of m into k sets for m<n.The Search AlgorithmThe search algorithm we used is the obvious branching scheme modified with various heuristics.We successively place an integer that is“most blocked”(in set membership)by earlier placements,and then backtrack when a branch dies or an answer is found.The heuristic used to decide which of the most blocked integers to select was to choose the smallest.Choosing the correct heuristic can make a huge difference in search ing this algorithm with the condition d(P)≥m,it is possible to completely exhaust the search tree in less than a minute for partitions of[1,n]into5sets with n near160and m near to44=S(4). For smaller depths the running time required to exhaust becomes excessive;but by limiting the total number of placements made beyond a certain level it is possible to probabilistically prune the search tree and have a good probability offinding an answer when one exists.Our search algorithm and heuristics were motivated by the following three observations that evolved during the development of the algorithm:1.It is easy to exhaustivelyfind all(not necessarily symmetric)sum-freepartitions into4sets.2.With high probability,it is easy to complete in a few steps a symmetricsum-free partition if only about20%of the elements are known.3.Many partitions are equivalent to partitions with large depth which canbe build byfirstfinding partitions in fewer sets.The partitions P of k=6and7sets given at the end of this paper were found using a probabilistic algorithm starting with a partition P of k−1sets with small d(P )and by restricting d(P)to be near S(k−1).the electronic journal of combinatorics7(2000),#R324 We believe that it should be possible tofind a larger lower bound for S(6)by looking for symmetric sum-free partitions that have large depth and have one other set with large e-depth.We currently have programs running to do this.Conjectures and Observations for5SetsThis section describes observations and conjections resulting from a computer study of symmetric sum-free partitions into5sets.By an exhaustive branching program,we were able to determine D5(n)for all n≤154.For102≤n≤154,D5(n)is always either44or45;for86≤n≤101, D5(n)is between41and45;and for45≤n≤85,D5(n)=[(n+1)/2].We were also able to determine that D5(160)=D5(157)=D5(156)=44,but that D5(159)=39.We were only able to show exhaustively that D5(155)<36and that D5(158)<36.A probabilistic search for symmetric sum-free partitions of 158into5parts,leads us to conjecture that there may be no such partitions; it is not obvious that a symmetric partition of n into k sets implies that there is a symmetric partition of m into k sets for m<n.A similar statement may apply to155,but we have not examined it as carefully.For160<n≤315 we exhaustively searched for symmetric partitions into5sets with depth≥44 and found that there are none.We did the same for other more relaxed depth restrictions,but for smaller n,with the same result;to summarizeD5(n)<36for161≤n≤190D5(n)<39for191≤n≤210D5(n)<44for211≤n≤315This,various probabilistic searches,and the upper bound in(3)leads us to conjecture that160is the largest integer for which there is a symmetric sum-free partition into5sets,and perhaps that S(5)=160.We found that there are symmetric sum-free partitions of157into5sets for each of the e-depths44,31,and26.For partitions of159into5sets,we found examples with D(P)=39and examples with D(P)=26.The only e-depth found for partitions of160into5sets was44.We conjecture that no other e-depths are possible for160into5sets.This would mean that all partitions of160’s would be equivalent to the5840we found exhaustively for 160with d=44;of these,768are equivalent to exactly one other partition with d(P)=44(interestingly,the equivalence multiplier is always30or59 (=−1/30mod161)),the rest were equivalent to no others with d(P)=44.So we conjecture that there are(4304+(768/2))∗φ(161)/2=309,408symmetric sum-free partitions of160into5sets.An interesting note for depth44symmetric sum-free partitions of160into 5sets is that the shortest distance between one and a multiple of another is small.Here the distance between partitions is defined as the smallest number of elements that are in different sets between the partitions,after a relabeling ofthe electronic journal of combinatorics7(2000),#R325 sets for one of the partitions.Considering only multiplication by1,30,and59 the maximum distance is19.This means that there is essentially only one cluster of depth44partitions,and if our conjecture is correct,then every symmetric partition is equivalent to something in this cluster.The situation in the last paragraph for160is similar to what happens for the largest partitions into4and3sets;further evidence that160may be maximal for5set partitions.For partitions of44into4sets all24=φ(45)symmetric sum-free partitions of the273sum-free partitions are equivalent(note that, since3divides45,multiplying by1and-1are not the same),and6of the24 have depth13.For sum-free partitions of13into3sets all three partitions are symmetric and equivalent.Another interesting aside,for depth44symmetric sum-free partitions of160 into5sets is that the sizes of the sets have the following properties:1.There are128different set size configurations.2.The largest set size is44,and the smallest is24.3.The largest difference between smallest and largest set in a partition is20,and the smallest is6.For the largest difference,the largest set is always44.For the smallest difference,the largest set is always34.4.The most likely set size configuration is{26,28,32,34,40},the electronic journal of combinatorics7(2000),#R326 ConstructionsThe constructions exhibited below are symmetric,and so we only list the small-est of a symmetric pair.Partition of536into6symmetric sumfree sets:Set1:1581114242730333640434649 526571778184909399103109112115125128131 134137144147150153160163166169172181185188191194 201204207213220223226229232235238242245248251254 264267358Set2:212192526344157586372798586 9596102118123124140141145146155156162173183193 200206211215216222233239244253260261266Set3:310162223293542485660626768 69747580878894100101106107113114119121126 133139151152158159164165171178184192197198203205 210217237241243249250255256Set4:4132028313850616473839198108 110117120132135142143154161168177179187195209212 214219221224231236246258265Set5:69172132394451545566708289 92104111127129130149167175189190202225227247252 262263Set6:715183745475359767897105116122 136138148157170174176180182186196199208218228230 234240257259268(D(P)=161)We also have constructions of6set partitions for the following n and respec-tive e-depths D:n=499,506,510,511,512,513,514,515,516,517,518,519,520,523,525,526,531 D=160,161,160,161,161,161,144,147,158,144,142,161,161,161,155,160,161.the electronic journal of combinatorics7(2000),#R327 Partition of1680into7symmetric sumfree sets:Set1:1151738525680828587899698109 129133135140142149151153158182184186190200206208 230237239241250253255261274283308318320334336352 371373389391415424428444446470477479486488493499 512521523525532541543546565567572574576583590596 627629643664666682684708717737750759761768770779 781800803810812814816821834Set2:22936425367738691100117124138147 155171188202209235240268280290291299306311317324 337350355361362383388393394421426432443459473476 481490506514528537544555561575588594599621626658 659665670681696697703714728740747752778785799832 Set3:38101221232830325061636870 7279818399101103105110112114121123130132134 141143150152154156161163165174176183185194196201 203205214221223232236238243245252254256258265267 272276278285287294296303305307314323325327329338 345347356358367374376380385387396398405407409420 427431438440447449456458460465467469471478480487 489491498507509511513518520522529531540542547551 558560562569573578580582591593600602609611613620 622624631633640644649662671673680691693695700706 711713715726731733735744746753757762773775777782 784793795797802806808813815817822824826828835837 Set4:41118203334356264657188115118 120167168170173217218220227242249270271273295297 300302309312326339348353370379406408411423425452 455461503505508533550564577605608617632646647661 663690699729741743780788789796811Set5:51439464749102104106136139189191192 199224233259262277284288315321340341342343344359 377403412429430437439441462474494496497515526527 538549556559579581597612614615623641650667668675 676678679694709712732734749764765767776794819820 829831Set6:69242526274143444555575859 74767792949597108111126127144145146159162 164177178179180193195197211212213215226229244246 247263264279282293330332333335346349364365366368the electronic journal of combinatorics7(2000),#R328 382384397399400402414416417418433434435448450451 453464466468482483484485500501502517534535536552 553568570571584585587603606618619634635637638652 653655656672685687688702704705718720721722723725 738755756758771772774787790791805809823825838840Set7:713161922313740485154606669 7578849093107113116119122125128131137148157 160166169172175181187198204207210216219222225228 231234248251257260266269275281286289292298301304 310313316319322328331351354357360363369372375378 381386390392395401404410413419422436442445454457 463472475492495504510516519524530539545548554557 563566586589592595598601604607610616625628630636 639642645648651654657660669674677683686689692698 701707710716719724727730736739742745748751754760 763766769783786792798801804807818827830833836839 (D(P)=537)We also have constructions of7set partitions forn=1615,1620,1630,1640,1650,1660,1665D=532,537,536,537,537,537,537. References[1]L.D.Baumert and S.W.Golomb.Backtrack p.Machinery,12:516–524,1965.[2]Albrecht Beutelspacher and Walter Brestovansky.Generalized Schur Num-bers.In Combinatorial Theory,Proceedings of a Conference Held at Schloss Rauischholzhausen,May6-9,1982,volume969of Lecture Notes in Mathe-matics,pages30–38.Springer-Verlag,Berlin-New York,1982.[3]Fan Rong K.Chung.On the Ramsey Numbers N(3,3,...3;2).DiscreteMathematics,5:317–321,1973.[4]Geoffrey Exoo.A Lower Bound for Schur Numbers and MulticolorRamsey Numbers of K3.The Electonic Journal of Combinatorics, /,#R8,1:3pages,1994.[5]Harold Fredricksen.Schur Numbers and the Ramsey NumbersN(3,3,...,3;2)binatorial Theory Ser A,27:371–379,1979.the electronic journal of combinatorics7(2000),#R329 [6]Stainislaw P.Radziszowski.Small Ramsey Numbers.Dynamic Survey DS1.Electronic binatorics,1:35pages,1984Revision#6July5,1999. [7]I.Schur.Uber die Kongruenz x m+y m≡z m(mod p).Jahresber.Deutsch.Math.-Verin.,25:114–116,1916.。