The Maximum Number of Subset Divisors of a Given Size

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1 The Maxiu Nuber of Subet Divior of a Give Size arxiv: v [ath.co] 0 May 05 Abtract Sauel Zbary Caregie Mello Uiverity a zbary@yahoo.co Matheatic Subject Claificatio: 05A5, 05D05 If i a poitive iteger ad A i a et of poitive iteger, we ay that B i a -divior of A if b B b a A a. We tudy the axial uber of -ubet of a -eleet et that ca be -divior. We provide a couterexaple to a cojecture of Huyh that for, the awer i ( with oly fiitely ay exceptio, but prove that addig a eceary coditio ae thi true. Moreover, we how that uder a iilar coditio, the awer i with oly fiitely ay exceptio for each. ( Itroductio If X i a et of poitive iteger, let X deote x X x. Let A be a fiite ubet of the poitive iteger. The eleet of A are a < a < < a ad let B be a ubet of A. We ay that B i a divior of A if B A. We defie d (A to be the uber of -ubet divior of A ad let d(, be the axiu value of d (A over all et A of poitive iteger. Siilarly, for a poitive iteger, we ay that B i a -divior of A if B A. We defie d (A to be the uber of -ubet -divior of A ad let d (, be the axiu value of d (A over all et A of poitive iteger. Note that the cocept of divior ad -divior coicide. Alo, if B i a divior of A, the B i a -divior of A for all, o d (A d (A ad d (, d(, Huyh [3] ote that for ay value of a,..., a, we ca pic uch a a that ay -ubet of {a,..., a } will be a A-divior. Therefore d(, ( for all. Thi ( otivate the defiitio that A i a -ati-pecil if the et of -ubet divior of A i A\{a}. We iilarly defie A to be a (, -ati-pecil if the et of -ubet -divior of A i ( A\{a }. Huyh [3] alo forulate the followig cojecture (Cojecture.

2 Cojecture. For all but fiitely ay value of ad, d(, (. I thi paper, we provide ifiite failie of couterexaple, but prove that, with the exceptio of thee failie, the cojecture i true. Thi give u the followig odified for. Cojecture. For all but fiitely ay iteger pair (, with < <, d(, (. For coveiece, we ow recale, dividig every eleet of A by A, o that ow the eleet of A are poitive ratioal uber ad A. Uder thi recalig, B A i a divior of A if ad oly if B for oe poitive iteger ad B i a -divior of A if ad oly if B for oe poitive iteger. Clearly, the value of d(, ad d (, do ot chage. The < coditio i Cojecture i eceary ice it i eay to ee that d(, > (. Alo, if { A, } 4,...,, 3(, 3( the A, o d (A ad d(, > (. Therefore the < coditio i eceary. However, we prove that thee failie cover all but fiitely ay exceptio. Theore 3. For all but fiitely ay pair (,, if < <, A, ad d ( (, the A i a -ati-pecil. Note that thi iediately iplie Cojecture. If we are itereted i -divior, we get aother faily of exceptio. If, a + ad a, the + d (A, o d (, > (. However, we prove that thee cover all but fiitely ay exceptio. Theore 4. Fix. For all but fiitely ay pair (, (with the uber of thee pair depedig o, if < <, A, ad d ( (, the A i a (, -ati-pecil. Note that thi iediately iplie the followig corollary. Corollary 5. Fix. The d (, ( for all but fiitely ay pair (, with < < (with the uber of thee pair depedig o. We will prove Theore 4. I the cae, where, if i, the (A\{ai } >, o A\{a i} i ot a divior of A. Thi, together with the cae of Theore 4, give u Theore 3. 3 Lea Tae a d-dieioal lattice cube with lattice poit per edge. Defie a poet o the lattice poit by (x,..., x d (y,..., y d if x i y i for all i.

3 Lea 6. The larget atichai i thi poet ha at ot ( + d d d eleet. Proof. Firt, we eed oe defiitio. The width of a poet i the ize of it larget atichai. If P i a fiite poet, we ay that P i raed if there exit a fuctio ρ : P Z atifyig ρ(y ρ(x + if y cover x i P (i.e. y > x, ad there i o z P with y > z > x. If ρ(x i, the x i aid to have ra i. Let P i deote the et of eleet of P of ra i. We ay P i ra-yetric ra-uiodal if there exit oe c Z with P i P i+ whe i < c ad P c i P i for all i Z. A raed poet P i called trogly Sperer if for ay poitive iteger, the larget ubet of P that ha o ( + -chai i the uio of the larget P i. Proctor, Sa, ad Sturtevat [6] prove that the cla of ra-yetric ra-uiodal trogly Sperer poet i cloed uder product. Sice a liear orderig of legth i ra-yetric ra-uiodal trogly Sperer, o i a product of d of the (the lattice cube. Ceter the cube o the origi by tralatio i R d. Let U be the et of eleet whoe coordiate u to 0. Sice the poet i ra-yetric ra-uiodal trogly Sperer, it width i at ot the ize of P c, which i U. For each y (y,..., y d U, let S y be the et of poit (x,..., x d with x i y i < for i d (ote that thi doe ot iclude the lat idex which lie o the hyperplae give by x + + x d 0. If y, z are ditict eleet of U, the S y ad S z are clearly dijoit. Alo, the projectio of S y oto the hyperplae give by x d 0 i a uit (d - dieioal hypercube, which ha volue. Thu the volue of S y i d ad the volue of y U S y i U d. O the other had, if (x,..., x d S y, the x i y i < for i d ad x d y d d i x i y i < (d. Thu (x,..., x d lie i the cube of edge legth ( + (d + d cetered at the origi. Therefore y U S y lie i the iterectio of a cube of edge legth + d with a hyperplae through it ceter (the origi. Ball [] how that the volue of the iterectio of a uit hypercube of arbitrary dieio with a hyperplae through it ceter i at ot. Therefore the volue of y U S y i at ot ( + d d, o U ( + d d d. Let X {x < < x } be ay et of poitive iteger. If B, C ( X d, the we ay that B C if we ca write B {b,..., b d } ad C {c,..., c d } with b i c i for all i d. Wheever we copare ubet of A, we will be uig thi partial order. Lea 7. Fix d >. For ufficietly large, the width of the partial order defied above i le tha ( X. d d 3

4 Proof. Let U be a axiu atichai of the partial order. Tae the partial order of X d, which coicide with the cube partial order. Let U {(y,..., y d X d {y,..., y d } U}. Note that thi ea, i particular, that all eleet of ay -tuple i U are ditict. If (y,..., y d, (z,..., z d U with (y,..., y d < (z,..., z d, the we get that {y i } {z i } ad d i y i < d i z i, o {y i } {z i }, o {y i } < {z i }, which i ipoible. Thu U i a atichai of X d of ize d! U ad The ( X d ( d give u U d! ( + d d d. U ( X d ( + d d d ( ( d +. For ufficietly large, ( +d d d+ <, o U ( < X. d d Let d( deote the uber of divior of. Lea 8. For ay poitive iteger, d( O(. Proof. There are fiitely ay prie p <, o there ut be oe cotat C uch that for ay p < ad ay poitive iteger, d(p + C(p. For p >, d(p + (p. Thu if j i p i i for ditict prie p i, the d( j i d (p i i C j i ( (p i i C O. Lea 9. Fix poitive iteger,, a, b. The for poitive iteger, the uber of pair of poitive iteger (x, y uch that a + b ad all three fractio are i lowet ter i x y at ot O(. Proof. Aue a + b. Let p gcd(, x, with tp ad x wp. The x y Lettig q gcd(w at, twp, we get b y a x w at. twp w at qb. ( For ay choice of, p, q, ( give at ot oe poible value of w, thu at ot oe value of x, ad thu at ot oe value of (x, y. The defiitio of q give u q p. The for a give, both p ad q are divior of, o by Lea 8 there are O( poible value for p ad O( value for q, o there are O( value for (p, q ad O( pair of uber (x, y. 4

5 4 Proof of Theore 4 Aue that A, d (A (, ad that A i ot a (, -ati-pecil. Note that the oe B a ha B, o ice <, we have a + <. We will ue thi i + all ( the cae below. Alo, the uber of -ubet of A that are ot -divior i at ot ( (. Rear 0. If B ad C are -ubet of A with B < C, the B < C. Note that if B 0 < B < < B are all divior of A ad B < /q, the B 0 < /(q +. Therefore if a B 0, the a < /(q +. Sice <, B < /, o we autoatically get that a < /( + Each of the ubectio below i a eparate cae. 4. all Fix ad let >>. For i,..., i, call the ordered -tuple (i,..., i repetitive if ot all etrie are ditict. Call it good if all etrie are ditict ad {a ij } i a -divior. Otherwie, call the ordered -tuple bad. We will firt retrict our attetio to -tuple where i. Aog thee, O( are repetitive. Alo, O( iclude both ad aog their copoet. Of the reaider, at ot (! ( are bad. Thu at leat /3 of the -tuple (i,..., i atifyig i are good. By the Pigeohole Priciple, there are oe value j,..., j with j uch that the chai {(, j,..., j,..., (, j,..., j } U ha at leat /3 good -tuple. Thi give u a chai of -ubet -divior of legth at leat /3. Thu a 3. Let B {a i i > ( 9 }. If ai B, the A i a i < a i + 9 a + a < a i o a i > ad a 6 i >. 6 Thu ay -divior that i a ubet of B ut u to oe >, o there are at 6 ot 6 ditict value that ca tae. Thu there are at ot 6 ditict value that a -divior that i a ubet of B ca u to. If D ( B ad r for oe poitive iteger, call D a r-te if there are at leat pair {x, y} B\D with (D {x, y} r. Call uch pair tail of D. If two tail of D are {x, y} ad {x, z}, the the u coditio give u y z, o tail of D are pairwie dijoit. Now let B 0 B. Note that B 0 >. A log a B 0 i, B 0 i ha at ot ( ubet which are ot -divior of A, o it ha at leat ( Bi -ubet that are -divior. Sice thee ( tae o at ot 6 value, there ut be oe poitive iteger i uch that at leat Bi -ubet of Bi u to r i i. 5

6 If we radoly chooe D i ( B i, the expected value for the uber of pair {x, y} B i \D i with ( (D i {x, y} r i i at leat Bi (. Thu we will chooe a Di ( uch that the uber of thee pair i at leat Bi (. Sice ( Bi ( 5 ( B i 5 (0 0000, 6 D i atifie the defiitio of a r i -te. Let B i B i \D i. The for i, D 0 i i a r i -te ad all the D i are dijoit. Sice the uber of -ubet of A which are ot -divior i le tha ( 0, we ow that there ut exit dijoit D i,..., D i uch that ay et coitig of oe eleet of each D ij will be a -divior. Note that i the cae, D i D i. Partitio j D i j ito uch et C,..., C. Let p /(. For j, we wat to chooe T j ,..., Tp j to be tail of D ij. We will chooe the for j, the for j, ad o o. Whe we chooe {T j l }, we will ae each of thee tail dijoit fro each of the te, a well a fro the already choe tail. Thi i poible ice D ih h j p h l T h l j p D ih + h h l T h l ( + (j p ( + ( p. Sice ay eleet i a te or i a previouly choe tail ca be i at ot oe tail of D ij, at ot ( + ( p tail are eliiated, o there ut be at leat p tail till available to chooe fro. We ay that a choice of tail {T j i j } j for each te i fortuitou if {x j i j } j ad {y j i j } j are both -divior. There are p > /( choice of tail, ad at ot ( of the are ot fortuitou. Thu at leat of poible choice are fortuitou. By the Pigeohole Priciple, we ca chooe i,..., i o that there are at leat p/ choice for i which ae {Ti,..., T i, Ti } fortuitou. Note that differet choice of i give u differet value of x i ad therefore differet value of j xj i j, o j xj i j ca tae o at leat p/ Ω( differet value. O the other had, if we are give a fortuitou choice of tail {T j i j }, the Cj + x j i j + y j i j j j x j i j + j j j y j i j j ( D ij {x j i j, y j i j } r ij Cj. j j 6

7 The right had ide doe ot deped o our choice of tail. Alo, ice each r ij ad each Cj ha deoiator at ot 6, the right had ide ha deoiator at ot (6. Sice both j xj i j ad j yj i j are -divior, there are at ot poibilitie for their uerator. For each uch poibility, by Lea 9, j xj i j ca tae o at ot ( (6 O ( 4 O differet value. Thu j xj i j ca tae o at ot ( O 3 differet value, cotradictig the upper boud above. 4. 3, ufficietly large Let d (( + /0.03. Aue that i ufficietly large relative d ad that 3. Let T be the et of -ubet of A that iclude both a ad a. Let T be the et of -ubet of A that iclude oe of a or a, but ot both. Defie U ad U iilarly, but with ( d-ubet. For S U t, let P S {B T t S B} (the et of -ubet obtaiable by addig d eleet of A le tha a to S. Note that a eleet of T t i cotaied i P S for exactly ( t d value of S. Thu if α Tt eleet of T t are -divior, the there i oe S U t o that at leat α P S eleet of P S are -divior. Now ote that the dijoit uio T T i the et of all -ubet whoe greatet eleet i at leat a, o T T ( ( 7

8 ad the fractio of the eleet of T T which are ot -divior i at ot ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 0.76 for ufficietly large. Therefore, if we et α 0.4, the for t or t, the fractio of eleet of T t that are -divior i at leat α, o for oe S, the fractio of eleet of P S which are -divior i at leat α 0.4. Note that the partial order of P S i the ae a the partial order of ( A\S\{a,a } d, o by Lea 7, it width i at ot P d S. The, by Miry theore, there i a chai of -ubet -divior i P S of legth at leat α P S P d S 0. d( (0.03 d. But the the firt eleet of the chai iclude a or a, o by Rear 0, a The a i 0.03 d ad, ice a < yieldig a cotradictio. i, + a i < < < ( 6 + 3, ufficietly large Let d ( Aue that i ufficietly large ad that 3 < < d. i (0.03. d 8

9 Radoly arrage the eleet of A aroud a circle. Let M be the et of -ubet of A coitig of coecutive eleet aroud the circle, ad let N {B M B }. (+ If B, C N ad they are hifted relative each other by at leat, the A\(B C, o A (A\(B C + B + C < + + ( + + ( +, which i ipoible. Thu ay two eleet of N are hifted by at ot aroud the circle. Thi give u N. Sice ay -ubet of A uig to at ot ha equal probability (+ of beig i N, thi tell u that the uber of -ubet with u at ot i at ot (+. Thu there are at leat ( ( ( ( ( ( -ubet which are -divior of A ad have a u of eleet greater tha. The u (+ of eleet of uch a et i >, o it ca tae o oe of ( + (+ + value, o there ut be oe iteger o that at leat ( +( of the -ubet of A u to (. Thu at leat ( + of the ( -ubet of A u to. If S ( A d, let PS be the et of ( -ubet obtaiable by addig d eleet of A to S. Note that ay ( -ubet of A i cotaied i P S for exactly ( d value of S, o there i oe S o that at leat ( + P S eleet of P S u to. They ut the for a atichai. However, the partial order of P S i the ae a the partial order of ( A\S d, o by Lea 7, it larget atichai ha ize le tha yieldig a cotradictio. d + d P S < 3 d P S ( + P S, 4.4 ( <, ufficietly large Aue that (6 + 3 <. Let u. Thu < u (6 + 3, o u ca tae o oly fiitely ay value. Aue that i ufficietly large relative thoe value. Let { ( A Y B A\B i a -divior of A}. u By auptio, Y ( ( u. 9

10 Let q be a all a poible o that a q <. Note that q < u( +. If B ( A u(+ u ad b a q for all b B, the B <, o (A\B > ad B / Y. Thu every + + B Y cotai at leat oe of the q greatet eleet of A. The uber of u-ubet of A cotaiig at leat of the q greatet eleet of A i bouded by ( ( q q < u(+ < Y, u u o at leat half of the eleet of Y cotai exactly oe of the q greatet eleet of A. Thu there ( ut be oe a i which i oe of the q greatet eleet of A uch that at leat u(+ u eleet of Y iclude ai ad o other of the q larget eleet. If B i uch a eleet of Y, the B < ai + (u u( + < + + u u( + u( +. Sice B ut be of the for for oe poitive iteger, we get fewer tha (+u poible value of. Thu there ut be oe value of o that there are at leat ( u ( + u differet u-ubet of A which iclude a i ad u to. However, if we have a collectio of that ay u-ubet of A that cotai a i, the oe of the will hare u eleet ad thu have differet u. Thi give u a cotradictio. 5 Cocluio For ufficietly large, all are covered by oe of the three lat cae. For all, all but fiitely value of are covered by the firt cae. I the tateet of Theore 3 ad Theore 4, all but fiitely ay caot be oitted. For exaple, Huyh[3] ote that 4,, A {, 5, 7, } give d (A 4 > (. A icreae, the uber of uch exceptio grow; i fact, it i eay to ee that ay,, A, will be a exceptio for ufficietly large. We could follow the proof ad trace out the upper boud o uch that (, i a exceptio; however thee will probably be far fro optial (for itace, for, (, 4 i liely the oly exceptio. It would be iteretig to get a good boud o the uber of uch exceptio, or o how large ca be i ter of for (, to be a exceptio. I thi paper, we are coutig B ( A uch that B. If we itead couted B uch that B <, thi proble becoe equivalet to the Maica-Miló-Sighi cojecture: Cojecture. For poitive iteger, with 4, every et of real uber with oegative u ha at leat ( -eleet ubet whoe u i alo oegative. 0

11 The equivalece i give by taig the copleet of B ad applyig a liear traforatio. The MMS cojecture ha bee prove for [4], 0 46 [5], ad 8 [], however there are pair (, uch that it doe ot hold. Thi ugget a ore geeral proble. Proble. Fix S [0, ] ad poitive iteger ad. If A i a et of poitive real, let d (S, A be the uber of ubet B ( A uch that B S. Let d(s,, be the axial value of d(s,, over all A with A ad A. For what S,, do we get d(s,, (? Furtherore, whe doe d (S, A ( iply that A i a -ati-pecil? Thi paper addree thi proble for S { Z+ }, while the MMS cojecture deal with thi proble for S (0, /. Aother exaple of a et for which thi proble ight be iteretig i a et of the for S (0, α/ { Z+ }, which cobie the theore of thi paper with the MMS cojecture. 6 Acowledgeet Thi reearch wa coducted a part of the Uiverity of Mieota Duluth REU progra, upported by NSA grat H ad NSF grat I would lie to tha Joe Gallia for hi advice ad upport. I would alo lie to tha Ada Heterberg ad Tiothy Chow for helpful dicuio ad uggetio. I would alo lie to tha Bria Scott for a ueful awer o Math Stac Exchage ( Referece [] K. Ball. Cube licig i R. Proc. Aer. Math. Soc., 97(3:465 47, 986. [] A. Chowdhury, G. Sari, ad S. Shahriari. A ew quadratic boud for the Maica- Miló-Sighi cojecture. 04. preprit, [3] Toy Huyh. Extreal proble for ubet divior. Elect. J. Cobi., :P.4, 04. [4] N. Maica ad D. Miló. O the uber of oegative partial u of a oegative u. I Cobiatoric (Eger, 987, Colloq. Math. Soc. Jáo Bolyai 5 (North-Hollad, 988, [5] A. Poroviy. A liear boud o the Maica-Milo-Sighi cojecture. 03. preprit, [6] R. A. Proctor, M. Sa, ad D. Sturtevat. Product partial order with the Sperer property. Dicrete Math., 30:73 80, 980.