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Zero-sum Ramsey theory

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615: 563: 471: 436: 169: 315: 377: 348: 140: 241: 215: 261: 189: 76: 56: 656: 58:), one seeks for conditions that guarantee the existence of certain substructure whose weights of its elements sum up to zero (in 272: 675: 268: 649: 264: 680: 441: 382: 145: 642: 594: 536: 579: 95: 575: 511: 482: 285: 106: 569: 515: 353: 324: 116: 220: 194: 626: 246: 174: 87: 61: 41: 669: 622: 102: 79: 36: 32: 28: 614: 502:
Erdős, Paul; Ginzburg, A.; Ziv, A. (1961). "Theorem in the additive number theory".
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structure whose elements are assigned different weights (usually elements from an
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summing to zero.) There are known proofs of this result using the
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is the order of the group) which adds to zero. It is known that
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Generalizing this result, one can define for any abelian group
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that sums to zero. (This bound is tight, as a sequence of
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The classic result in this area is the 1961 theorem of
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Zero-sum trees: a survey of results and open problems
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Finite and Infinite Combinatorics in Sets and Logic
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Study of structures where a subset must sum to zero
570:Zero-Sum Ramsey Theory: Graphs, Sequences and More 465: 430: 371: 342: 309: 255: 235: 209: 183: 163: 134: 70: 50: 438:, and that this bound is strict if and only if 650: 8: 657: 643: 457: 453: 452: 443: 384: 355: 326: 321:such that there must be a subsequence of 287: 248: 222: 196: 176: 155: 151: 150: 147: 118: 63: 43: 494: 590:, Kluwer Acad. Publ. (1993) pp. 19–29 98:, and other branches of mathematics. 7: 611: 609: 531: 529: 527: 525: 243:ones cannot have any subset of size 466:{\displaystyle G=\mathbb {Z} _{m}} 431:{\displaystyle EGZ(G)\leq 2o(G)-1} 14: 613: 164:{\displaystyle \mathbb {Z} _{m}} 419: 413: 401: 395: 366: 360: 337: 331: 304: 298: 1: 566:(open-access journal article) 629:. You can help Knowledge by 564:Zero-sum problems - A survey 171:, there is a subset of size 595:Zero-sum problems: a survey 697: 608: 78:). It combines tools from 504:Bull. Res. Council Israel 273:Chevalley–Warning theorem 265:Cauchy-Davenport theorem 269:Fermat's little theorem 625:-related article is a 541:encyclopediaofmath.org 467: 432: 373: 344: 311: 310:{\displaystyle EGZ(G)} 257: 237: 211: 185: 165: 136: 72: 52: 21:zero-sum Ramsey theory 468: 433: 374: 345: 312: 282:the minimum quantity 258: 238: 212: 186: 166: 137: 73: 53: 442: 383: 372:{\displaystyle o(G)} 354: 343:{\displaystyle o(G)} 325: 286: 247: 221: 195: 175: 146: 135:{\displaystyle 2m-1} 117: 62: 42: 676:Combinatorics stubs 572:(workshop homepage) 236:{\displaystyle m-1} 210:{\displaystyle m-1} 463: 428: 369: 340: 307: 253: 233: 207: 181: 161: 132: 68: 48: 638: 637: 605:(1996) pp. 93–113 256:{\displaystyle m} 184:{\displaystyle m} 96:discrete analysis 71:{\displaystyle A} 51:{\displaystyle A} 688: 659: 652: 645: 617: 610: 576:Arie Bialostocki 551: 550: 548: 547: 533: 520: 519: 499: 483:Zero-sum problem 472: 470: 469: 464: 462: 461: 456: 437: 435: 434: 429: 378: 376: 375: 370: 350:elements (where 349: 347: 346: 341: 316: 314: 313: 308: 262: 260: 259: 254: 242: 240: 239: 234: 216: 214: 213: 208: 190: 188: 187: 182: 170: 168: 167: 162: 160: 159: 154: 141: 139: 138: 133: 107:Abraham Ginzburg 77: 75: 74: 69: 57: 55: 54: 49: 19:In mathematics, 696: 695: 691: 690: 689: 687: 686: 685: 666: 665: 664: 663: 560: 558:Further reading 555: 554: 545: 543: 535: 534: 523: 501: 500: 496: 491: 479: 451: 440: 439: 381: 380: 352: 351: 323: 322: 317:of elements of 284: 283: 245: 244: 219: 218: 193: 192: 173: 172: 149: 144: 143: 115: 114: 60: 59: 40: 39: 27:is a branch of 25:zero-sum theory 17: 12: 11: 5: 694: 692: 684: 683: 678: 668: 667: 662: 661: 654: 647: 639: 636: 635: 618: 607: 606: 599:Discrete Math. 591: 573: 567: 559: 556: 553: 552: 521: 493: 492: 490: 487: 486: 485: 478: 475: 460: 455: 450: 447: 427: 424: 421: 418: 415: 412: 409: 406: 403: 400: 397: 394: 391: 388: 368: 365: 362: 359: 339: 336: 333: 330: 306: 303: 300: 297: 294: 291: 252: 232: 229: 226: 206: 203: 200: 180: 158: 153: 131: 128: 125: 122: 88:linear algebra 67: 47: 15: 13: 10: 9: 6: 4: 3: 2: 693: 682: 681:Ramsey theory 679: 677: 674: 673: 671: 660: 655: 653: 648: 646: 641: 640: 634: 632: 628: 624: 623:combinatorics 619: 616: 612: 604: 600: 596: 592: 589: 588:Nato ASI Ser. 585: 581: 577: 574: 571: 568: 565: 562: 561: 557: 542: 538: 532: 530: 528: 526: 522: 517: 513: 509: 505: 498: 495: 488: 484: 481: 480: 476: 474: 458: 448: 445: 425: 422: 416: 410: 407: 404: 398: 392: 389: 386: 363: 357: 334: 328: 320: 301: 295: 292: 289: 281: 276: 274: 270: 266: 250: 230: 227: 224: 204: 201: 198: 178: 156: 129: 126: 123: 120: 112: 108: 104: 99: 97: 93: 89: 85: 81: 80:number theory 65: 45: 38: 37:Abelian group 34: 33:combinatorial 30: 29:combinatorics 26: 22: 631:expanding it 620: 602: 598: 587: 583: 544:. Retrieved 540: 507: 503: 497: 318: 279: 277: 142:elements of 100: 92:graph theory 24: 20: 18: 217:zeroes and 111:Abraham Ziv 670:Categories 593:Y. Caro, " 546:2023-05-22 516:0063.00009 489:References 113:: for any 103:Paul ErdĹ‘s 510:: 41–43. 423:− 405:≤ 271:, or the 228:− 202:− 127:− 477:See also 84:algebra 514:  109:, and 621:This 627:stub 603:152 578:, " 512:Zbl 508:10F 23:or 672:: 601:, 597:" 586:, 539:. 524:^ 506:. 473:. 275:. 267:, 105:, 94:, 90:, 86:, 82:, 658:e 651:t 644:v 633:. 549:. 518:. 459:m 454:Z 449:= 446:G 426:1 420:) 417:G 414:( 411:o 408:2 402:) 399:G 396:( 393:Z 390:G 387:E 367:) 364:G 361:( 358:o 338:) 335:G 332:( 329:o 319:G 305:) 302:G 299:( 296:Z 293:G 290:E 280:G 251:m 231:1 225:m 205:1 199:m 179:m 157:m 152:Z 130:1 124:m 121:2 66:A 46:A

Index

combinatorics
combinatorial
Abelian group
number theory
algebra
linear algebra
graph theory
discrete analysis
Paul Erdős
Abraham Ginzburg
Abraham Ziv
Cauchy-Davenport theorem
Fermat's little theorem
Chevalley–Warning theorem
Zero-sum problem
Zbl
0063.00009




"Erdös-Ginzburg-Ziv theorem - Encyclopedia of Mathematics"
Zero-sum problems - A survey
Zero-Sum Ramsey Theory: Graphs, Sequences and More
Arie Bialostocki
Zero-sum trees: a survey of results and open problems
Zero-sum problems: a survey
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combinatorics
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